37,562 research outputs found

    Entanglement in a second order topological insulator on a square lattice

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    In a dd-dimensional topological insulator of order dd, there are zero energy states on its corners which have close relationship with its entanglement behaviors. We studied the bipartite entanglement spectra for different subsystem shapes and found that only when the entanglement boundary has corners matching the lattice, exact zero modes exist in the entanglement spectrum corresponding to the zero energy states caused by the same physical corners. We then considered finite size systems in which case these corner states are coupled together by long range hybridizations to form a multipartite entangled state. We proposed a scheme to calculate the quadripartite entanglement entropy on the square lattice, which is well described by a four-sites toy model and thus provides another way to identify the higher order topological insulators from the multipartite entanglement point of view.Comment: 5 pages, 3 figure

    Cyclotomy and permutation polynomials of large indices

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    We use cyclotomy to design new classes of permutation polynomials over finite fields. This allows us to generate many classes of permutation polynomials in an algorithmic way. Many of them are permutation polynomials of large indices

    Promotion operator on rigged configurations of type A

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    Recently, the analogue of the promotion operator on crystals of type A under a generalization of the bijection of Kerov, Kirillov and Reshetikhin between crystals (or Littlewood--Richardson tableaux) and rigged configurations was proposed. In this paper, we give a proof of this conjecture. This shows in particular that the bijection between tensor products of type A_n^{(1)} crystals and (unrestricted) rigged configurations is an affine crystal isomorphism.Comment: 37 page

    Promotion and evacuation on standard Young tableaux of rectangle and staircase shape

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    (Dual-)promotion and (dual-)evacuation are bijections on SYT(\lambda) for any partition \lambda. Let c^r denote the rectangular partition (c,...,c) of height r, and let sc_k (k > 2) denote the staircase partition (k,k-1,...,1). B. Rhoades showed representation-theoretically that promotion on SYT(c^r) exhibits the cyclic sieving phenomenon (CSP). In this paper, we demonstrate a promotion- and evacuation-preserving embedding of SYT(sc_k) into SYT(k^{k+1}). This arose from an attempt to demonstrate the CSP of promotion action on SYT(sc_k).Comment: 14 pages, typos correcte

    On the number of NN-free elements with prescribed trace

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    In this paper we derive a formula for the number of NN-free elements over a finite field Fq\mathbb{F}_q with prescribed trace, in particular trace zero, in terms of Gaussian periods. As a consequence, we derive a simple explicit formula for the number of primitive elements, in quartic extensions of Mersenne prime fields, having absolute trace zero. We also give a simple formula in the case when Q=(qm−1)/(q−1)Q = (q^m-1)/(q-1) is prime. More generally, for a positive integer NN whose prime factors divide QQ and satisfy the so called semi-primitive condition, we give an explicit formula for the number of NN-free elements with arbitrary trace. In addition we show that if all the prime factors of q−1q-1 divide mm, then the number of primitive elements in Fqm\mathbb{F}_{q^m}, with prescribed non-zero trace, is uniformly distributed. Finally we explore the related number, Pq,m,N(c)P_{q, m, N}(c), of elements in Fqm\mathbb{F}_{q^m} with multiplicative order NN and having trace c∈Fqc \in \mathbb{F}_q. Let N∣qm−1N \mid q^m-1 such that LQ∣NL_Q \mid N, where LQL_Q is the largest factor of qm−1q^m-1 with the same radical as that of QQ. We show there exists an element in Fqm∗\mathbb{F}_{q^m}^* of (large) order NN with trace 00 if and only if m≠2m \neq 2 and (q,m)≠(4,3)(q,m) \neq (4,3). Moreover we derive an explicit formula for the number of elements in Fp4\mathbb{F}_{p^4} with the corresponding large order LQ=2(p+1)(p2+1)L_Q = 2(p+1)(p^2+1) and having absolute trace zero, where pp is a Mersenne prime

    Modelling and Simulations of Multi-component Lipid Membranes and Open Membranes via Diffusive Interface Approaches

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    In this paper, phase field models are developed for multi-component vesicle membranes with different lipid compositions and membranes with free boundary. These models are used to simulate the deformation of membranes under the elastic bending energy and the line tension energy with prescribed volume and surface area constraints. By comparing our numerical simulations with recent experiments, it is demonstrated that the phase field models can capture the rich phenomena associated with the membrane transformation, thus it offers great functionality in the simulation and modeling of multicomponent membranes

    A probabilistic approach to value sets of polynomials over finite fields

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    In this paper we study the distribution of the size of the value set for a random polynomial with degree at most q−1q-1 over a finite field Fq\mathbb{F}_q. We obtain the exact probability distribution and show that the number of missing values tends to a normal distribution as qq goes to infinity. We obtain these results through a study of a random rr-th order cyclotomic mappings. A variation on the size of the union of some random sets is also considered
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