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Europhysics Letters (EPL)

Quantum Coherence in an Exactly Solvable One-Dimensional Model with Defects

Peter Schmitteckert, P. Schwab, Ulrich Eckern

Published Jun 20, 1995• 54 citations
Publisher

Abstract

Using the Quantum Inverse Scattering Method we construct an integrable Heisenberg-XXZ model, or equivalently a model for spinless fermions with nearest-neighbour interaction, with defects. Each defect involves three sites with a fine tuning between nearest-neighbour and next-nearest-neighbour terms. We investigate the finite-size corrections to the ground-state energy and its dependence on an external flux as a function of a parameter ν, characterizing the strength of the defects. For intermediate values of ν, both quantities become very small, although the ground-state wave function remains extended.

DOI
10.1209/0295-5075/30/9/007

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Quantum Coherence in an Exactly Solvable One-dimensional Model with Defects

Peter Schmitteckert, P. Schwab, U. Eckern

Abstract. – Using the Quantum Inverse Scattering Method we construct an integrable Heisenberg-XXZ-model, or equivalently a model for spinless fermions with nearest-neighbour interaction, with defects. Each defect involves three sites with a fine tuning between nearest-neighbour and nextnearest-neighbour terms. We investigate the finite size corrections to the ground state energy and its dependence on an external flux as a function of a parameter ν, characterizing the strength of the defects. For intermediate values of ν, both quantities become very small, although the ground state wavefunction remains extended. ( ∗) Electronic mail:

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Reply to the comment by A.A. Zvyagin and H. Johannesson on our paper “Quantum coherence in an exactly solvable one-dimensional model with defects”

Peter Schmitteckert, P. Schwab, Ulrich Eckern

The comment [CITE] by Zvyagin and Johannesson (ZJ) discusses some of the issues raised in our recent letter [CITE] in which we construct an exactly solvable one-dimensional quantum model, which corresponds to a Heisenberg- XXZ model or spinless fermions on a ring, with defects. In particular, we presented results for the finite-size corrections to the ground-state energy as well as for its flux sensitivity, as a function of a parameter ν which characterizes the strength of the defects.

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We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.

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Using the density-matrix renormalization-group algorithm, we study the model of spinless fermions with nearest-neighbor interaction on a ring in the presence of disorder. We determine the spatial decay of the density induced by a defect (Friedel oscillations), and the phase sensitivity of the ground-state energy \ensuremath{\Delta}E=(-${)}^{\mathit{N}}$[E(\ensuremath{\varphi}=0)-E(\ensuremath{\varphi}=\ensuremath{\pi})], where \ensuremath{\varphi}=2\ensuremath{\pi}\ensuremath{\Phi}/${\mathrm{\ensuremath{\Phi}}}_{0}$ (N is the number of fermions, \ensuremath{\Phi} the magnetic flux, and ${\mathrm{\ensuremath{\Phi}}}_{0}$=h/e the flux quantum), for a disordered system versus the system size M. The quantity ln(M\ensuremath{\Delta}E) is found to have a normal distribution to a good approximation. The ``localization length'' decreases (increases) for a repulsive (attractive) interaction. \textcopyright{} 1996 The American Physical Society.

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Comment on “Quantum coherence in an exactly solvable one-dimensional model with defects”

Andrei A. Zvyagin, Henrik Johannesson

In a recent paper by Schmitteckert et al. , an exactly solvable XXZ chain with defects was studied. We here point out that, for this type of model, the critical exponents for the asymptotic correlation functions are insensitive to the presence of defects. Also, the defects do not enhance the persistent current of the equivalent fermionic model, contrary to a claim by Schmitteckert et al.

Europhysics Letters (EPL)•Jul 10, 1996•7 citations

Integrable spin-1/2 XXZ Heisenberg chain with competing interactions

Holger Frahm

The critical behaviour of an integrable model of a spin- 1/2 chain with nearest-neighbour XXZ interaction and a competing three-spin interaction involving nearest and next-nearest neighbours is studied. The phase diagram at zero temperature is obtained. Methods from conformal field theory are used to compute the asymptotics of the spin-spin correlation functions.

Journal of Physics A Mathematical and General•Mar 21, 1992•56 citations
Using the Quantum Inverse Scattering Method we construct an integrable Heisenberg-XXZ model, or equivalently a model for spinless fermions with nearest-neighbour interaction, with defects. Each defect involves three sites with a fine tuning between nearest-neighbour and next-nearest-neighbour terms. We investigate the finite-size corrections to the ground-state energy and its dependence on an external flux as a function of a parameter ν, characterizing the strength of the defects. For intermediate values of ν, both quantities become very small, although the ground-state wave function remains extended.
Abstract. – Using the Quantum Inverse Scattering Method we construct an integrable Heisenberg-XXZ-model, or equivalently a model for spinless fermions with nearest-neighbour interaction, with defects. Each defect involves three sites with a fine tuning between nearest-neighbour and nextnearest-neighbour terms. We investigate the finite size corrections to the ground state energy and its dependence on an external flux as a function of a parameter ν, characterizing the strength of the defects. For intermediate values of ν, both quantities become very small, although the ground state wavefunction remains extended. ( ∗) Electronic mail:
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
Using the density-matrix renormalization-group algorithm, we study the model of spinless fermions with nearest-neighbor interaction on a ring in the presence of disorder. We determine the spatial decay of the density induced by a defect (Friedel oscillations), and the phase sensitivity of the ground-state energy \ensuremath{\Delta}E=(-)N{)}^{\mathit{N}})N[E(\ensuremath{\varphi}=0)-E(\ensuremath{\varphi}=\ensuremath{\pi})], where \ensuremath{\varphi}=2\ensuremath{\pi}\ensuremath{\Phi}/${\mathrm{\ensuremath{\Phi}}}_{0}$ (N is the number of fermions, \ensuremath{\Phi} the magnetic flux, and ${\mathrm{\ensuremath{\Phi}}}_{0}$=h/e the flux quantum), for a disordered system versus the system size M. The quantity ln(M\ensuremath{\Delta}E) is found to have a normal distribution to a good approximation. The ``localization length'' decreases (increases) for a repulsive (attractive) interaction. \textcopyright{} 1996 The American Physical Society.