Browsing by Subject "lattice"

Now showing 1 - 4 of 4
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    Biclosed sets in Combinatorics
    (2015-08) McConville, Thomas
    The weak order is the set of permutations of [n] partially ordered by inclusion of inversion sets. This partial order arises naturally in various contexts, including enumer- ative combinatorics, hyperplane arrangements, Schubert calculus, cluster algebras, and many more. A fundamental result on the weak order is that the collection of maximal chains in any interval is connected by certain "local moves"�. Other notable features of the weak order are its lattice structure, its topology, and its geometry. The collection of inversion sets of permutations is an example of a family of biclosed sets. This thesis focuses on extending various nice properties of the weak order to other posets of biclosed sets. Some of these collections of biclosed sets have appeared previously in the literature, while others seem to be new. We briefly summarize our main results below. (�3.1.3) We give a criterion on a closure operator which ensures that the poset of biclosed sets is a congruence-uniform lattice. (�4) The chambers of a real simplicial or supersolvable hyperplane arrangement are in natural bijection with biclosed subsets of hyperplanes. (�4) completing the proof in [81]) The graph of reduced galleries of a supersolv- able hyperplane arrangement has diameter equal to the number of codimension 2 intersection subspaces. (�5) Chamber posets are semidistributive lattices if and only if they are crosscut- simplicial if and only if the arrangement is bineighborly. (�6) Every interval of the second Higher Bruhat order is either contractible or homotopy equivalent to a sphere. (�7) Every "facial"� interval of the poset of reduced galleries of a supersolvable arrangement is homotopy equivalent to a sphere. (�8) The Grid-Tamari orders are congruence-uniform lattices.
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    The Flexibility of the Six Vertex Lattice Model in the Study of Special Functions
    (2022-06) Frechette, Claire
    In this thesis, we examine the six-vertex lattice model and three generalizations thereof, whose partition functions give three different kinds of special functions: double biaxial (β,q)-Grothendieck polynomials, supersymmetric LLT functions, and metaplectic spherical Whittaker functions. Modelling these functions on a solvable lattice allows us to prove functional equations and identities by using the Yang-Baxter equations associated to the lattice model. Lattice models also encode an immense amount of data from the underlying structure of a space of special functions, and we will examine how different interpretations of this data visualize different properties of the functions, including fundamental connections to quantum groups.
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    MATLAB Code: Raman Scattering Intensity for Quadratic Hamiltonians
    (2015-09-14) Perreault, Brent M; perre035@umn.edu; Perreault, Brent
    These are codes were used to generate the Raman scattering intensity spectra of Kitaev Spin Liquid models using the Loudon-Fleury approach. In its most basic form this code diagonalizes a quadratic fermionic Hamiltonian and computes the spectra by constructing the Raman operator, using the eigenfunctions to compute matrix elements, and the eigenvalues to plot the spectrum as a function of energy. Variants are included that consider 2D and 3D lattices, finite systems, as well as the resonant Raman scattering. Neither interactions nor the bosonic case are considered.
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    A new approach to lattice quantum field theories
    (2018-01) Jain, Mudit
    In this thesis report, I describe an algorithm for lattice simulation of quantum/statistical fields that reduces the complexity of current techniques (Metropolis algorithm) from exponential in all the directions of space and (Euclidean-)time, to linear in (Euclidean-)time and exponential in space. This is done by building a typical field configuration spatial slice by spatial slice through an analytically obtained Markov chain from its path integral. Although the complexity still depends exponentially on the number of spatial lattice points, for quantum mechanics ($0+1$ fields) spatial slice is only a point and thus the complexity only depends linearly on the number of time lattice points and simulation becomes pretty easy. As examples, I discuss the cases of harmonic and an-harmonic oscillators along with some simulation results. The case of Gaussian fields in general (in any dimension) is trivial since in the similarity transformed space each lattice site decouples and hence there exists a random variable at each lattice site that does not interact with any other. Although the reduction of complexity from exponential in space (if possible) for higher dimensional fields in general is currently under investigation, I present a checkerboard network that we investigated along with some simulation results.