arXiv subjects
Publications and source records attributed to Xi Chen.
With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix of the eight-vertex solid-on-solid (SOS) model, we obtain the determinant representations of the scalar products of Bethe states for the open XYZ chain with non-diagonal boundary terms. By taking the on shell limit, we obtain the determinant representations (or Gaudin formula) of the norms of the Bethe states.
With the help of the Drinfeld twist or factorizing F-matrix for the eight-vertex solid-on-solid (SOS) model, we find that in the F-basis provided by the twist the two sets of pseudo-particle creation operators simultaneously take completely symmetric and polarization free form. This allows us to obtain the explicit and completely symmetric expressions of the two sets of Bethe states of the model.
Insulating substrates are crucial for electrical transport study and room temperature application of topological insulator films at thickness of only several nanometers. High quality quantum well films of Bi2Se3, a typical three-dimensional topological insulator, have been grown on α-Al2O3 (sapphire) (0001) by molecular beam epitaxy. The films exhibit well-defined quantum well states and surface states, suggesting the uniform thickness over macroscopic area. The Bi2Se3 thin films on sapphire (0001) provide a good system to study low-dimensional physics of topological insulators since conduction contribution from the substrate is negligibly small.
We prove a complexity dichotomy theorem for all non-negative weighted counting Constraint Satisfaction Problems (CSP). This caps a long series of important results on counting problems including unweighted and weighted graph homomorphisms and the celebrated dichotomy theorem for unweighted #CSP. Our dichotomy theorem gives a succinct criterion for tractability. If a set F of constraint functions satisfies the criterion, then the counting CSP problem defined by F is solvable in polynomial time; if it does not satisfy the criterion, then the problem is #P-hard. We furthermore show that the question of whether F satisfies the criterion is decidable in NP. Surprisingly, our tractability criterion is simpler than the previous criteria for the more restricted classes of problems, although when specialized to those cases, they are logically equivalent. Our proof mainly uses Linear Algebra, and represents a departure from Universal Algebra, the dominant methodology in recent years.
With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix for the open XXZ spin chain with non-diagonal boundary terms, we obtain the determinant representations of the scalar products of Bethe states of the model.
Any polarization vector of a plane wave can be decomposed into a pair of mutually orthogonal base vectors, known as a polarization basis. Regarding this decomposition as a quasi-unitary transformation from a three-component vector to a corresponding two-component spinor, one is led to a representation formalism for the photon spin. The spin operator $\hat{\boldsymbol γ}$ defined on the space of unit spinors, referred to as the Jones space, has only component along the wave vector and is represented by one of the Pauli matrices in the commonly used polarization basis. It is deformed by the quasi-unitary transformation from the spin operator that is defined on the space of unit polarization vectors, referred to as the Pancharatnam space. On the basis of this theory, it is shown that the Cartesian components of spin operator $\hat{\boldsymbol γ}$ are mutually commutative and the spin angular momentum in units of $\hbar$ is exactly the component of the Stokes vector along the wave vector.
We report a transport study of ultrathin Bi2Se3 topological insulators with thickness from one quintuple layer to six quintuple layers grown by molecular beam epitaxy. At low temperatures, the film resistance increases logarithmically with decreasing temperature, revealing an insulating ground state. The sharp increase of resistance with magnetic field, however, indicates the existence of weak antilocalization, which should reduce the resistance as temperature decreases. We show that these apparently contradictory behaviors can be understood by considering the electron interaction effect, which plays a crucial role in determining the electronic ground state of topological insulators in the two dimensional limit.
We use the dynamical invariants associated with the Hamiltonian of an atom in a one dimensional moving trap to inverse engineer the trap motion and perform fast atomic transport without final vibrational heating. The atom is driven non-adiabatically through a shortcut to the result of adiabatic, slow trap motion. For harmonic potentials this only requires designing appropriate trap trajectories, whereas perfect transport in anharmonic traps may be achieved by applying an extra field to compensate the forces in the rest frame of the trap. The results can be extended to atom stopping or launching. The limitations due to geometrical constraints, energies and accelerations involved are analyzed, as well as the relation to previous approaches (based on classical trajectories or "fast-forward" and "bang-bang" methods) which can be integrated in the invariant-based framework.
There is recently a surge of interest to cut down the time it takes to change the state of a quantum system adiabatically. We study for the time-dependent harmonic oscillator the transient energy excitation in speed-up processes designed to reproduce the initial populations at some predetermined final frequency and time, providing lower bounds and examples. Implications for the limits imposed to the process times and for the principle of unattainability of the absolute zero, in a single expansion or in quantum refrigerator cycles, are drawn.
In this paper, inspired by the work of Megiddo on the formation of preferences and strategic analysis, we consider an early market model studied in the field of economic theory, in which each trader's utility may be influenced by the bundles of goods obtained by her social neighbors. The goal of this paper is to understand and characterize the impact of social influence on the complexity of computing and approximating market equilibria. We present complexity-theoretic and algorithmic results for approximating market equilibria in this model with focus on two concrete influence models based on the traditional linear utility functions. Recall that an Arrow-Debreu market equilibrium in a conventional exchange market with linear utility functions can be computed in polynomial time by convex programming. Our complexity results show that even a bounded-degree, planar influence network can significantly increase the difficulty of equilibrium computation even in markets with only a constant number of goods. Our algorithmic results suggest that finding an approximate equilibrium in markets with hierarchical influence networks might be easier than that in markets with arbitrary neighborhood structures. By demonstrating a simple market with a constant number of goods and a bounded-degree, planar influence graph whose equilibrium is PPAD-hard to approximate, we also provide a counterexample to a common belief, which we refer to as the myth of a constant number of goods, that equilibria in markets with a constant number of goods are easy to compute or easy to approximate.
We prove that a very general projective K3 surface does not admit a dominant self rational map of degree at least two.
Bi2Se3 is theoretically predicted1 2and experimentally observed2,3 to be a three dimensional topological insulator. For possible applications, it is important to understand the electronic structure of the planar device. In this work, thickness dependent band structure of molecular beam epitaxy grown ultrathin films of Bi2Se3 is investigated by in situ angle-resolved photoemission spectroscopy. An energy gap is observed for the first time in the topologically protected metallic surface states of bulk Bi2Se3 below the thickness of six quintuple layers, due to the coupling between the surface states from two opposite surfaces of the Bi2Se3 film. The gapped surface states exhibit sizable Rashba-type spin-orbit splitting, due to breaking of structural inversion symmetry induced by SiC substrate. The spin-splitting can be controlled by tuning the potential difference between the two surfaces.
The complexity of graph homomorphism problems has been the subject of intense study. It is a long standing open problem to give a (decidable) complexity dichotomy theorem for the partition function of directed graph homomorphisms. In this paper, we prove a decidable complexity dichotomy theorem for this problem and our theorem applies to all non-negative weighted form of the problem: given any fixed matrix A with non-negative algebraic entries, the partition function Z_A(G) of directed graph homomorphisms from any directed graph G is either tractable in polynomial time or #P-hard, depending on the matrix A. The proof of the dichotomy theorem is combinatorial, but involves the definition of an infinite family of graph homomorphism problems. The proof of its decidability is algebraic using properties of polynomials.
In this paper, the electronic band structures and its transport properties in the gapped graphene superlattices, with one-dimensional (1D) periodic potentials of square barriers, are systematically investigated. It is found that a zero averaged wave-number (zero-$\overline{k}$ ) gap is formed inside the gapped graphene-based superlattices, and the condition for obtaining such a zero-$\overline{k}$ gap is analytically presented. The properties of this zero-$\overline{k}$ gap including its transmission, conductance and Fano factor are studied in detail. Finally it is revealed that the properties of the electronic transmission, conductance and Fano factor near the zero-$\overline{k}$ gap are very insensitive to the structural disorder for the finite graphene-based periodic-barrier systems.
Atomically flat thin films of topological insulator Bi2Se3 have been grown on double-layer graphene formed on 6H-SiC(0001) substrate by molecular beam epitaxy. By a combined study of reflection high energy electron diffraction and scanning tunneling microscopy, we identified the Se-rich condition and temperature criterion for layer-by-layer growth of epitaxial Bi2Se3 films. The as-grown films without doping exhibit a low defect density of 1.0\pm 0.2x1011/cm2, and become a bulk insulator at a thickness of 10 quintuple layers, as revealed by in situ angle resolved photoemission spectroscopy measurement.
Undirected graphical models encode in a graph $G$ the dependency structure of a random vector $Y$. In many applications, it is of interest to model $Y$ given another random vector $X$ as input. We refer to the problem of estimating the graph $G(x)$ of $Y$ conditioned on $X=x$ as ``graph-valued regression.'' In this paper, we propose a semiparametric method for estimating $G(x)$ that builds a tree on the $X$ space just as in CART (classification and regression trees), but at each leaf of the tree estimates a graph. We call the method ``Graph-optimized CART,'' or Go-CART. We study the theoretical properties of Go-CART using dyadic partitioning trees, establishing oracle inequalities on risk minimization and tree partition consistency. We also demonstrate the application of Go-CART to a meteorological dataset, showing how graph-valued regression can provide a useful tool for analyzing complex data.
We consider the problem of learning a structured multi-task regression, where the output consists of multiple responses that are related by a graph and the correlated response variables are dependent on the common inputs in a sparse but synergistic manner. Previous methods such as l1/l2-regularized multi-task regression assume that all of the output variables are equally related to the inputs, although in many real-world problems, outputs are related in a complex manner. In this paper, we propose graph-guided fused lasso (GFlasso) for structured multi-task regression that exploits the graph structure over the output variables. We introduce a novel penalty function based on fusion penalty to encourage highly correlated outputs to share a common set of relevant inputs. In addition, we propose a simple yet efficient proximal-gradient method for optimizing GFlasso that can also be applied to any optimization problems with a convex smooth loss and the general class of fusion penalty defined on arbitrary graph structures. By exploiting the structure of the non-smooth ''fusion penalty'', our method achieves a faster convergence rate than the standard first-order method, sub-gradient method, and is significantly more scalable than the widely adopted second-order cone-programming and quadratic-programming formulations. In addition, we provide an analysis of the consistency property of the GFlasso model. Experimental results not only demonstrate the superiority of GFlasso over the standard lasso but also show the efficiency and scalability of our proximal-gradient method.
We consider the problem of evaluating certain exponential sums. These sums take the form $\sum_{x_1,...,x_n \in Z_N} e^{f(x_1,...,x_n) {2 πi / N}} $, where each x_i is summed over a ring Z_N, and f(x_1,...,x_n) is a multivariate polynomial with integer coefficients. We show that the sum can be evaluated in polynomial time in n and log N when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree \ge 3, we show the problem is #P-hard, even for any fixed prime or prime power modulus. This leads to a complexity dichotomy theorem - a complete classification of each problem to be either computable in polynomial time or #P-hard - for a class of exponential sums. These sums arise in the classifications of graph homomorphisms and some other counting CSP type problems, and these results lead to complexity dichotomy theorems. For the polynomial-time algorithm, Gauss sums form the basic building blocks. For the hardness results, we prove group-theoretic necessary conditions for tractability. These tests imply that the problem is #P-hard for even very restricted families of simple cubic polynomials over fixed modulus N.