The XX--model with boundaries. Part I: Diagonalization of the finite chain

This is the first of three papers dealing with the XX finite quantum chain with arbitrary, not necessarily hermitian, boundary terms. This extends previous work where the periodic or diagonal boundary terms were considered. In order to find the spectrum and wave-functions an auxiliary quantum chain is examined which is quadratic in fermionic creation and annihilation operators and hence diagonalizable. The secular equation is in general complicated but several cases were found when it can be solved analytically. For these cases the ground-state energies are given. The appearance of boundary states is also discussed and in view to the applications considered in the next papers, the one and two-point functions are expressed in terms of Pfaffians.Comment: 56 pages, LaTeX, some minor correction

Finite size scaling for quantum criticality using the finite-element method

Finite size scaling for the Schr\"{o}dinger equation is a systematic approach to calculate the quantum critical parameters for a given Hamiltonian. This approach has been shown to give very accurate results for critical parameters by using a systematic expansion with global basis-type functions. Recently, the finite element method was shown to be a powerful numerical method for ab initio electronic structure calculations with a variable real-space resolution. In this work, we demonstrate how to obtain quantum critical parameters by combining the finite element method (FEM) with finite size scaling (FSS) using different ab initio approximations and exact formulations. The critical parameters could be atomic nuclear charges, internuclear distances, electron density, disorder, lattice structure, and external fields for stability of atomic, molecular systems and quantum phase transitions of extended systems. To illustrate the effectiveness of this approach we provide detailed calculations of applying FEM to approximate solutions for the two-electron atom with varying nuclear charge; these include Hartree-Fock, density functional theory under the local density approximation, and an "exact"' formulation using FEM. We then use the FSS approach to determine its critical nuclear charge for stability; here, the size of the system is related to the number of elements used in the calculations. Results prove to be in good agreement with previous Slater-basis set calculations and demonstrate that it is possible to combine finite size scaling with the finite-element method by using ab initio calculations to obtain quantum critical parameters. The combined approach provides a promising first-principles approach to describe quantum phase transitions for materials and extended systems.Comment: 15 pages, 19 figures, revision based on suggestions by referee, accepted in Phys. Rev.

Asymmetric XXZ chain at the antiferromagnetic transition: Spectra and partition functions

The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of the spectra of the asymmetric XXZ chain and the accompanying isotropic 6-vertex model near the antiferromagnetic phase boundary at zero vertical field. The energy gaps scale with size $N$ as $N^{-1/2}$ and its amplitudes are obtained in terms of level-dependent scaling functions. Exactly on the phase boundary, the amplitudes are proportional to a sum of square-root of integers and an anomaly term. By summing over all low-lying levels, the partition functions are obtained explicitly. Similar analysis is performed also at the phase boundary of zero horizontal field in which case the energy gaps scale as $N^{-2}$. The partition functions for this case are found to be that of a nonrelativistic free fermion system. From symmetry of the lattice model under $\pi /2$ rotation, several identities between the partition functions are found. The $N^{-1/2}$ scaling at zero vertical field is interpreted as a feature arising from viewing the Pokrovsky-Talapov transition with the space and time coordinates interchanged.Comment: Minor corrections only. 18 pages in RevTex, 2 PS figure

Spectra of non-hermitian quantum spin chains describing boundary induced phase transitions

The spectrum of the non-hermitian asymmetric XXZ-chain with additional non-diagonal boundary terms is studied. The lowest lying eigenvalues are determined numerically. For the ferromagnetic and completely asymmetric chain that corresponds to a reaction-diffusion model with input and outflow of particles the smallest energy gap which corresponds directly to the inverse of the temporal correlation length shows the same properties as the spatial correlation length of the stationary state. For the antiferromagnetic chain with both boundary terms, we find a conformal invariant spectrum where the partition function corresponds to the one of a Coulomb gas with only magnetic charges shifted by a purely imaginary and a lattice-length dependent constant. Similar results are obtained by studying a toy model that can be diagonalized analytically in terms of free fermions.Comment: LaTeX, 26 pages, 1 figure, uses ioplppt.st

Some Exact Results for the Exclusion Process

The asymmetric simple exclusion process (ASEP) is a paradigm for non-equilibrium physics that appears as a building block to model various low-dimensional transport phenomena, ranging from intracellular traffic to quantum dots. We review some recent results obtained for the system on a periodic ring by using the Bethe Ansatz. We show that this method allows to derive analytically many properties of the dynamics of the model such as the spectral gap and the generating function of the current. We also discuss the solution of a generalized exclusion process with $N$-species of particles and explain how a geometric construction inspired from queuing theory sheds light on the Matrix Product Representation technique that has been very fruitful to derive exact results for the ASEP.Comment: 21 pages; Proceedings of STATPHYS24 (Cairns, Australia, July 2010

Generalized matrix Ansatz in the multispecies exclusion process - partially asymmetric case

We investigate one of the simplest multispecies generalization of the asymmetric simple exclusion process on a ring. This process has a rich combinatorial spectral structure and a matrix product form for the stationary state. In the totally asymmetric case operators that conjugate the dynamics of systems with different numbers of species were obtained by the authors and reported recently. The existence of such nontrivial operators was reformulated as a representation problem for a specific quadratic algebra (generalized matrix Ansatz). In the present work, we construct the family of representations explicitly for the partially asymmetric case. This solution cannot be obtained by a simple deformation of the totally asymmetric case