Entangled multi-knot lattice model of anyon current

We proposed an entangled multi-knot lattice model to explore the exotic statistics of anyon. This knot lattice model bears abelian and non-abelian anyons as well as integral and fractional filling states that is similar to quantum Hall system. The fusion rules of anyon are explicitly demonstrated by braiding on crossing states of the multi-knot lattice. The statistical character of anyon is quantified by topological linking number of multi-knot link. Long-range coupling interaction is a fundamental character of this knot lattice model. The short range coupling models, such as Ising model, fermion paring model, Kitaev honeycomb lattice model and so on, appears as the short range coupling case of the knot lattice model. We introduced link lattice pattern as geometric representation of the eigenstate of quantum many body model to explore the topological nature of quantum eigen-states. For example, a convection flow loop is introduced into the well-know BCS fermions pairing model to show the Pseudo-gap state in unconventional super-conducting state. The integral and fractional filling numbers in quantum Hall system is directly quantized by topological linking number. The quantum phase transition between different quantum states in quantum spin model is also directly quantified by the change of topological linking number, which revealed topological character of phase transition. This multi-knot lattice has a promising physical implementation by circularized photons in optical firbre network. It may also provide another different path to topological quantum computation.Comment: 32 pages, 30 figure

Topological Theory of Classical and Quantum Phase Transition

We presented the topological current of Ehrenfest definition of phase transition. It is shown that different topology of the configuration space corresponds to different phase transition, it is marked by the Euler number of the interaction potential. The two phases separated by the coexistence curve is assigned with different winding numbers of opposite sign. We also found an universal equation of coexistence curve, from which one can arrive the phase diagram of any order classical and quantum phase transition. The topological quantum phase transition theory is established, and is applied to the Bose-Hubbard model, the phase diagram of the first order quantum PT is in agreement with recent progress.Comment: 6 pages, added references for section

Odd-even effect of melting finite polymer film on square lattice

Two dimensional film system bears many exotic thermodynamics behaviors. We proposed a mathematical physics model to explore how the melting temperature of a two dimensional mathematical dimer film depends on the odd-eveness of the finite width of dimer film. A weak external bond between dimers is introduced into the classical dimer model in this dimer film. We derived a general equation of melting temperature and applied it for computing the melting temperature of a dimer film covering a finite square lattice. The melting temperature is proportional to the external bonding energy that we assume it binds neighboring dimers together and proportional to the inverse of entropy per site. Further more, it shows fusing two small rectangular dimer film with odd number of length into one big rectangular film gains more entropy than fusing two small rectangles with even number of length into the same big rectangle. Fusing two small toruses with even number of length into one big torus reduces entropy. Fusing two small toruses with odd number of length increases the entropy. Thus two dimer films with even number of length repel each other, two dimer films with odd length attract each other. The odd-even effect is also reflected on the correlation function of two topologically distinguishable loops in a torus surface. The entropy of finite system dominates odd-even effect. This model has straightforward extension to longer polymers and three dimensional systems.Comment: 10 pages in two column, 15 figures in Int. J. Mod. Phys. B 201

Convective Boson-Fermion pairing model constructed by oscillating one-dimensional optical superlattice

Boson-fermion mixture exist in nature as quark-gluon plasma and $^3$He-$^4$He mixture. We proposed a convective boson-fermion pairing theory, that can be implemented by ultracold atoms in optical superlattice transformation between different configurations. This transformation may induce the collision and division between boson and fermion, which defines a theoretical convective pairing state. The paring Hamiltonian is Hermitian but it always generate a complex energy spectrum. Each finite gap state can be classified by a topological winding number. The stable pairing state only exists for certain discrete momentum vector zones. An unstable linear dispersion connects two neighboring stable pairing states. The boson-fermion gap function controls the momentum gap space between two neighboring pairing state. The critical temperature of transition from a gapped to gapless phase shows a maximal value at negative fermion chemical potential. The density of state for the pairing excitation diverges at low energy, thus most pairing states are observable at low energy.Comment: 5 pages, 4 figure

Equation of motion for density distribution of many circling particles with an overdamped circle center

We first established the dynamic equations to describe the noisy circling motion of a single particle and the corresponding probability conservation equation in both two dimensions and three dimensions, and then developed the evolution equation of density distribution of many circling particles with overdamped circle center. For many circling particle system without any external force, the density gradient in one direction can induce a flow perpendicular to this direction. While for single circling particle, similar phenomena occurs only for non-zero external force. We performed numerical evolution of the density distribution of many circling particles, the density distribution behaves as a decaying Gaussian distribution propagating along the channel. We computed the particle flow field and the effective force field. Vortex shows up in the high density region. The force field drive particles to the transverse direction perpendicular to the density gradient. We applied this non-equilibrium evolution equation to understand the diffusion phenomena of many sperms(J. Exp. Biol. 210, 3805-3820). Numerical evolution gave us similar density distribution as experimental measurement. The transverse flow we predicted provide a theoretical understanding to the bias concentration of many sperms(J. Exp. Biol. 210. 3805-3820).Comment: 22 pages, added equations and arguments, added reference

Game Theory and Topological Phase Transition

Phase transition is a war game. It widely exists in different kinds of complex system beyond physics. Where there is revolution, there is phase transition. The renormalization group transformation, which was proved to be a powerful tool to study the critical phenomena, is actually a game process. The phase boundary between the old phase and new phase is the outcome of many rounds of negotiation between the old force and new force. The order of phase transition is determined by the cutoff of renormalization group transformation. This definition unified Ehrenfest's definition of phase transition in thermodynamic physics. If the strategy manifold has nontrivial topology, the topological relation would put a constrain on the surviving strategies, the transition occurred under this constrain may be called a topological one. If the strategy manifold is open and noncompact, phase transition is simply a game process, there is no table for topology. An universal phase coexistence equation is found, it sits at the Nash equilibrium point. Inspired by the fractal space structure demonstrated by renormalization group theory, a conjecture is proposed that the universal scaling law of a general phase transition in a complex system comes from the coexistence equation around Nash equilibrium point. Game theory also provide us new understanding to pairing mechanism and entanglement in many body physics.Comment: Revised version, 43 pages, 7 figures, Grammar mistake are corrected. Detail explanations are adde

The Hall effect of dipole chain in one dimensional Bose-Einstein condensation

We find a breather behavior of the dipole chain, this breather excitation obey fractional statistics, it could be an experimental quantity to detect anyon. A Hall effect of magnetic monopole in a dipole chain of ultracold molecules is also presented, we show that this Hall effect can induce the flip of magnetic dipole chain.Comment: 4 page

Exactly soluble spin-1/2 models on three-dimensional lattices and non-abelian statistics of closed string excitations

Exactly soluble spin-$\frac{1}2$ models on three-dimensional lattices are proposed by generalizing Kitaev model on honeycomb lattice to three dimensions with proper periodic boundary conditions. The simplest example is spins on a diamond lattice which is exactly soluble. The ground state sector of the model may be mapped into a p-wave paired state on cubic lattice. We observe for the first time a topological phase transition from a gapless phase to a gapped phase in an exactly soluble spin model. Furthermore, the gapless phase can not be gapped by a perturbation breaking the time reversal symmetry. Unknotted and unlinked Wilson loops arise as eigen excitations, which may evolute into linked and knotted loop excitations. We show that these closed string excitations obey abelian statistics in the gapped phase and non-abelian statistics in the gapless phase.Comment: 4 pages, 4 figures,references adde

Explicit demonstration of nonabelian anyon, braiding matrix and fusion rules in the Kitaev-type spin honeycomb lattice models

The exact solubility of the Kitaev-type spin honeycomb lattice model was proved by means of a Majorana fermion representation or a Jordan-Wigner transformation while the explicit form of the anyon in terms of Pauli matrices became not transparent. The nonabelian statistics of anyons and the fusion rules can only be expressed in indirect ways to Pauli matrices. We convert the ground state and anyonic excitations back to the forms of Pauli matrices and explicitly demonstrate the nonabelian anyonic statistics as well as the fusion rules. These results may instruct the experimental realization of the nonabelian anyons. We suggest a proof-in-principle experiment to verify the existence of the nonabelian anyons in nature.Comment: 4 pages, 2 figures; references added; some discussions on the A phase added; derivations to the fusion rules and braiding matrix are refine

Explicit illustration of non-abelian fusion rules in a small spin lattice

We exactly solve a four-site spin model with site-dependent Kitaev's coupling in a tetrahedron by means of an analytical diagonalization. The non-abelian fusion rules of eigen vortex excitations in this small lattice model are explicitly illustrated in real space by using Pauli matrices. Comparing with solutions of Kitaev models on large lattices, our solution gives an intuitional picture using real space spin configurations to directly express zero modes of Majorana fermions, non-abelian vortices and non-abelian fusion rules. We generalize the single tetrahedron model to a chain model of tetrahedrons on a torus and find the non-abelian vortices become well-defined non-abelian anyons. We believe these manifest results are very helpful to demonstrate the nonabelian anyon in laboratory.Comment: 4 pages, 2 figures, typos in eq.(13) corrected, reference adde