Transfer Matrices for the Partition Function of the Potts Model on Toroidal Lattice Strips

We present a method for calculating transfer matrices for the $q$-state Potts model partition functions $Z(G,q,v)$, for arbitrary $q$ and temperature variable $v$, on strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices of width $L_y$ vertices and of arbitrarily great length $L_x$ vertices, subject to toroidal and Klein bottle boundary conditions. For the toroidal case we express the partition function as $Z(\Lambda, L_y \times L_x,q,v) = \sum_{d=0}^{L_y} \sum_j b_j^{(d)} (\lambda_{Z,\Lambda,L_y,d,j})^m$, where $\Lambda$ denotes lattice type, $b_j^{(d)}$ are specified polynomials of degree $d$ in $q$, $\lambda_{Z,\Lambda,L_y,d,j}$ are eigenvalues of the transfer matrix $T_{Z,\Lambda,L_y,d}$ in the degree-$d$ subspace, and $m=L_x$ ($L_x/2$) for $\Lambda=sq, tri (hc)$, respectively. An analogous formula is given for Klein bottle strips. We exhibit a method for calculating $T_{Z,\Lambda,L_y,d}$ for arbitrary $L_y$. In particular, we find some very simple formulas for the determinant $det(T_{Z,\Lambda,L_y,d})$, and trace $Tr(T_{Z,\Lambda,L_y})$. Corresponding results are given for the equivalent Tutte polynomials for these lattice strips and illustrative examples are included.Comment: 52 pages, latex, 10 figure

Community detection for networks with unipartite and bipartite structure

Finding community structures in networks is important in network science, technology, and applications. To date, most algorithms that aim to find community structures only focus either on unipartite or bipartite networks. A unipartite network consists of one set of nodes and a bipartite network consists of two nonoverlapping sets of nodes with only links joining the nodes in different sets. However, a third type of network exists, defined here as the mixture network. Just like a bipartite network, a mixture network also consists of two sets of nodes, but some nodes may simultaneously belong to two sets, which breaks the nonoverlapping restriction of a bipartite network. The mixture network can be considered as a general case, with unipartite and bipartite networks viewed as its limiting cases. A mixture network can represent not only all the unipartite and bipartite networks, but also a wide range of real-world networks that cannot be properly represented as either unipartite or bipartite networks in fields such as biology and social science. Based on this observation, we first propose a probabilistic model that can find modules in unipartite, bipartite, and mixture networks in a unified framework based on the link community model for a unipartite undirected network [B Ball et al (2011 Phys. Rev. E 84 036103)]. We test our algorithm on synthetic networks (both overlapping and nonoverlapping communities) and apply it to two real-world networks: a southern women bipartite network and a human transcriptional regulatory mixture network. The results suggest that our model performs well for all three types of networks, is competitive with other algorithms for unipartite or bipartite networks, and is applicable to real-world networks.Comment: 27 pages, 8 figures. (http://iopscience.iop.org/1367-2630/16/9/093001

On Exactly Solvable Potentials

We investigate two methods of obtaining exactly solvable potentials with analytic forms.Comment: 13 pages, Latex, to appear in Chineses Journal of Physic

Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips

We present transfer matrices for the zero-temperature partition function of the $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial) on cyclic and M\"obius strips of the square, triangular, and honeycomb lattices of width $L_y$ and arbitrarily great length $L_x$. We relate these results to our earlier exact solutions for square-lattice strips with $L_y=3,4,5$, triangular-lattice strips with $L_y=2,3,4$, and honeycomb-lattice strips with $L_y=2,3$ and periodic or twisted periodic boundary conditions. We give a general expression for the chromatic polynomial of a M\"obius strip of a lattice $\Lambda$ and exact results for a subset of honeycomb-lattice transfer matrices, both of which are valid for arbitrary strip width $L_y$. New results are presented for the $L_y=5$ strip of the triangular lattice and the $L_y=4$ and $L_y=5$ strips of the honeycomb lattice. Using these results and taking the infinite-length limit $L_x \to \infty$, we determine the continuous accumulation locus of the zeros of the above partition function in the complex $q$ plane, including the maximal real point of nonanalyticity of the degeneracy per site, $W$ as a function of $q$.Comment: 62 pages, latex, 6 eps figures, includes additional results, e.g., loci ${\cal B}$, requested by refere

Electron Electric Dipole Moment induced by Octet-Colored Scalars

An appended sector of two octet-colored scalars, each an electroweak doublet, is an interesting extension of the simple two Higgs doublet model motivated by the minimal flavor violation. Their rich CP violating interaction gives rise to a sizable electron electric dipole moment, besides the quark electric dipole moment via the two-loop contribution of Barr-Zee mechanism.Comment: 8 pages, 2 figure

Chiral Restoration in the Early Universe: Pion Halo in the Sky

vanishing above $T_c$ indicates chiral symmetry restoration at high $T$. But is it the old $T=0$ chiral symmetry that is `restored'? In this talk, I report on the spacetime quantization of the BPFTW effective action for quarks in a hot environ. The fermion propagator is known to give a pseudo-Lorentz invariant particle pole as well as new spacelike cuts. Our quantization shows that the spacelike cuts directly lead to a thermal vacuum that is a generalized NJL state, with a curious $90^{o}$ phase. This $90^{o}$ is responsible for vanishing at high $T$. The thermal vacuum is invariant under a new chiral charge, but continues to break the old zero temperature chirality. Our quantization suggests a new class of order parameters that probe the physics of these spacelike cuts. In usual scenario, the pion dissociates in the early alphabet soup. With this new understanding of the thermal vacuum, the pion remains a Nambu-Goldstone particle at high $T$, and will not dissociate. It propagates at the speed of light but with a halo.Comment: 4 pages, LaTeX, CCNY-HEP-94-9 To appear in Proceedings of "Trends in Astroparticle Physics Workshop", Stockholm, Sweden, 22-25 September, 1994, Nuclear Physics B, Proceedings Supplement, edited by L. Bergstrom, P. Carlson, P.O. Hulth, and H. Snellman. (Only revision is in the header citation

Secure secret sharing in the cloud

In this paper, we show how a dealer with limited resources is possible to share the secrets to players via an untrusted cloud server without compromising the privacy of the secrets. This scheme permits a batch of two secret messages to be shared to two players in such a way that the secrets are reconstructable if and only if two of them collaborate. An individual share reveals absolutely no information about the secrets to the player. The secret messages are obfuscated by encryption and thus give no information to the cloud server. Furthermore, the scheme is compatible with the Paillier cryptosystem and other cryptosystems of the same type. In light of the recent developments in privacy-preserving watermarking technology, we further model the proposed scheme as a variant of reversible watermarking in the encrypted domain