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Register a new userDiscrete Holomorphic Parafermions in the Eight Vertex Model
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We show that holomorphic Parafermions exist in the eight vertex model. This is done by extending the definition from the six vertex model to the eight vertex model utilizing a parameter redefinition. These Parafermions exist on the critical plane and integrable cases of the eight vertex model. We show that for the case of staggered eight vertex model, these Parafermions correspond to those of the Ashkin-Teller model. Furthermore, the loop representation of the eight vertex model enabled us to show a connection with the O(n) model which is in agreement with the six vertex limit found as a special case of the O(n) model.
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We present the numbers of ice model and eight-vertex model configurations (with Boltzmann factors equal to one), I(n) and E(n) respectively, on the two-dimensional Sierpinski gasket SG(n) at stage $n$. For the eight-vertex model, the number of configurations is $E(n)=2^{3(3^n+1)/2}$ and the entropy per site, defined as $lim_{v to infty} ln E(n)/v$ where $v$ is the number of vertices on SG(n), is exactly equal to $ln 2$. For the ice model, the upper and lower bounds for the entropy per site $lim_{v to infty} ln I(n)/v$ are derived in terms of the results at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accurate. The corresponding result of ice model on the generalized two-dimensional Sierpinski gasket SG_b(n) with $b=3$ is also obtained. For the generalized vertex model on SG_3(n), the number of configurations is $2^{(8 times 6^n +7)/5}$ and the entropy per site is equal to $frac87 ln 2$. The general upper and lower bounds for the entropy per site for arbitrary $b$ are conjectured.
The spectrum of the quantum Ising chain can be found by expressing the spins in terms of free fermions. An analogous transformation exists for clock chains with $Z_n$ symmetry, but is of less use because the resulting parafermionic operators remain interacting. Nonetheless, Baxter showed that a certain non-hermitian (but PT-symmetric) clock Hamiltonian is free, in the sense that the entire spectrum is found in terms of independent energy levels, with the striking feature that there are $n$ possibilities for occupying each level. Here I show this directly explicitly finding shift operators obeying a $Z_n$ generalization of the Clifford algebra. I also find higher Hamiltonians that commute with Baxters and prove their spectrum comes from the same set of energy levels. This thus provides an explicit notion of a free parafermion. A byproduct is an elegant method for the solution of the Ising/Kitaev chain with spatially varying couplings.
We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models provide an efficient integrable Trotterization of non-relativistic $sigma$-models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau-Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar-Parisi-Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.
We study numerically the two-point correlation functions of height functions in the six-vertex model with domain wall boundary conditions. The correlation functions and the height functions are computed by the Markov chain Monte-Carlo algorithm. Particular attention is paid to the free fermionic point ($Delta=0$), for which the correlation functions are obtained analytically in the thermodynamic limit. A good agreement of the exact and numerical results for the free fermionic point allows us to extend calculations to the disordered ($|Delta|<1$) phase and to monitor the logarithm-like behavior of correlation functions there. For the antiferroelectric ($Delta<-1$) phase, the exponential decrease of correlation functions is observed.
We construct lattice parafermions - local products of order and disorder operators - in nearest-neighbor Z(N) models on regular isotropic planar lattices, and show that they are discretely holomorphic, that is they satisfy discrete Cauchy-Riemann equations, precisely at the critical Fateev-Zamolodchikov (FZ) integrable points. We generalize our analysis to models with anisotropic interactions, showing that, as long as the lattice is correctly embedded in the plane, such discretely holomorphic parafermions exist for particular values of the couplings which we identify as the anisotropic FZ points. These results extend to more general inhomogeneous lattice models as long as the covering lattice admits a rhombic embedding in the plane.
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