Bethe roots and refined enumeration of alternating-sign matrices

The properties of the most probable ground state candidate for the XXZ spin chain with the anisotropy parameter equal to -1/2 and an odd number of sites is considered. Some linear combinations of the components of the considered state, divided by the maximal component, coincide with the elementary symmetric polynomials in the corresponding Bethe roots. It is proved that those polynomials are equal to the numbers providing the refined enumeration of the alternating-sign matrices of order M+1 divided by the total number of the alternating-sign matrices of order M, for the chain of length 2M+1.Comment: LaTeX 2e, 12 pages, minor corrections, references adde

Direct SIMS Determination of the InxGa1-xN Mole Fraction

We demonstrate that our secondary mass ion spectroscopy (SIMS) method for the determination of the mole fraction in solid InxGa1-xN solutions is accurate and reproduceable without need of reference samples. The method is based on measuring relative current values of CsM+ (M=Ga, In) secondary ions. The claim of reliable SIMS determination without reference samples was confirmed by four independent analytical methods on the same samples with a relative error in the InN mole fraction determination below 15

Exact expressions for correlations in the ground state of the dense O(1) loop model

Conjectures for analytical expressions for correlations in the dense O$(1)$ loop model on semi infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the existing results. We also consider boundaries on which loops can end. We call such boundaries ''open''. We have obtained expressions for correlations when both boundaries are open, and one is open and the other one is reflecting. Also, we formulate a conjecture relating the ground state of the model with open boundaries to Fully Packed Loop models on a finite square grid. We also review earlier obtained results about this relation for the three other types of boundary conditions. Finally, we construct a mapping between the ground state of the dense O$(1)$ loop model and the XXZ spin chain for the different types of boundary conditions.Comment: 25 pages, version accepted by JSTA

Refined Razumov-Stroganov conjectures for open boundaries

Recently it has been conjectured that the ground-state of a Markovian Hamiltonian, with one boundary operator, acting in a link pattern space is related to vertically and horizontally symmetric alternating-sign matrices (equivalently fully-packed loop configurations (FPL) on a grid with special boundaries).We extend this conjecture by introducing an arbitrary boundary parameter. We show that the parameter dependent ground state is related to refined vertically symmetric alternating-sign matrices i.e. with prescribed configurations (respectively, prescribed FPL configurations) in the next to central row. We also conjecture a relation between the ground-state of a Markovian Hamiltonian with two boundary operators and arbitrary coefficients and some doubly refined (dependence on two parameters) FPL configurations. Our conjectures might be useful in the study of ground-states of the O(1) and XXZ models, as well as the stationary states of Raise and Peel models.Comment: 11 pages LaTeX, 8 postscript figure

On FPL configurations with four sets of nested arches

The problem of counting the number of Fully Packed Loop (FPL) configurations with four sets of a,b,c,d nested arches is addressed. It is shown that it may be expressed as the problem of enumeration of tilings of a domain of the triangular lattice with a conic singularity. After reexpression in terms of non-intersecting lines, the Lindstr\"om-Gessel-Viennot theorem leads to a formula as a sum of determinants. This is made quite explicit when min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates the numbers of configurations with b=d.Comment: 22 pages, TeX, 16 figures; a new formula for a generating function adde

Non-local space-time supersymmetry on the lattice

We show that several well-known one-dimensional quantum systems possess a hidden nonlocal supersymmetry. The simplest example is the open XXZ spin chain with \Delta=-1/2. We use the supersymmetry to place lower bounds on the ground state energy with various boundary conditions. For an odd number of sites in the periodic chain, and with a particular boundary magnetic field in the open chain, we can derive the ground state energy exactly. The supersymmetry thus explains why it is possible to solve the Bethe equations for the ground state in these cases. We also show that a similar space-time supersymmetry holds for the t-J model at its integrable ferromagnetic point, where the space-time supersymmetry and the Hamiltonian it yields coexist with a global u(1|2) graded Lie algebra symmetry. Possible generalizations to other algebras are discussed.Comment: 12 page

Baxter operators for the quantum sl(3) invariant spin chain

The noncompact homogeneous sl(3) invariant spin chains are considered. We show that the transfer matrix with generic auxiliary space is factorized into the product of three sl(3) invariant commuting operators. These operators satisfy the finite difference equations in the spectral parameters which follow from the structure of the reducible sl(3) modules.Comment: 20 pages, 4 figures, references adde

Entanglement and correlation in anisotropic quantum spin systems

Analytical expressions for the entanglement measures concurrence, i-concurrence and 3-tangle in terms of spin correlation functions are derived using general symmetries of the quantum spin system. These relations are exploited for the one-dimensional XXZ-model, in particular the concurrence and the critical temperature for disentanglement are calculated for finite systems with up to six qubits. A recent NMR quantum error correction experiment is analyzed within the framework of the proposed theoretical approach.Comment: 8 pages, 3 figure