Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1908.09698 · Journal of Statistical Mechanics Theory and Experiment

The interaction energy of the particle-hole pair in the XXZ -spin chain

A A Ovchinnikov

Published Sep 1, 2019• 1 citation
arXivHTMLPDF

Abstract

Abstract We directly calculate the interaction energy of an eigenstate with a single particle and a single hole for the XXZ -spin chain. We find the connection of this addition to energy with a two-particle forward scattering phase shift function. The results can be applied to the calculation of the exponents of the threshold singularities in the correlators of the XXZ -spin chain.

DOI
10.1088/1742-5468/ab3aee

Related papers

Threshold singularities in the XXZ-spin chain

Alexander A. Ovchinnikov

We calculate the critical exponents of the threshold singularity for the spectral density of the XXZ-spin chain at zero magnetic field for the lower threshold. We show that the corresponding phase shifts are momentum independent and coincide with predictions of the effective mobile impurity Hamiltonian approach.

arXiv:2004.11371•Modern Physics Letters B•Oct 14, 2020•1 citation

Exact Edge Singularities and Dynamical Correlations in Spin- 1 / 2 Chains

Rodrigo G. Pereira, Steven R. White, Ian K. Affleck

Exact formulas for the singularities of the dynamical structure factor, Szz(q,omega), of the S=1/2 xxz spin chain at all q and any anisotropy and magnetic field in the critical regime are derived, expressing the exponents in terms of the phase shifts which are known exactly from the Bethe ansatz solution. We also study the long-time asymptotics of the self-correlation function 0|Sjz(t)Sjz(0)|0. Utilizing these results to supplement very accurate time-dependent density matrix renormalization group, for short to moderate times, we calculate Szz(q,omega) to very high precision.

arXiv:0709.0960•Physical Review Letters•Jan 17, 2008•133 citations

Spin 1/2 Fermions in the Unitary Limit

H. S. Køhler

The energy shift due to the interaction of two particles in a large box is proportional to the free particle scattering phase-shift. This provides an approximation to the effective interaction referred to as the phase-shift approximation. For a many-body fermion system this effective interaction has to be corrected for the Pauli-blocking. It is used to calculate the energy of a spin 1/2 fermion system as a function of the two-body scattering length and effective range. In the unitary limit with scattering length going to infinity and effective range going to zero the energy is 0.540 (in the units of the non-interactiong fermi gas energy) with a pp-ladder summation. Including hh-ladders the energy is 0.570. A smooth crossover from the BCS to the BEC region is observed. Pairing is not included.

arXiv:nucl-th/0701006•Jan 2, 2007•0 citations

Study of the ground state of the one-dimensional Heisenberg spin-1/2 chain

K.R. de Ruiter

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.

•Jan 1, 2018•0 citations

On singularities of dynamic response functions in the massless regime of the XXZ spin-1/2 chain

Karol Kajetan Kozlowski

This work extracts, by means of an exact analysis, the singular behavior of the dynamical response functions—the Fourier transforms of dynamical two-point functions—in the vicinity of the various excitation thresholds in the massless regime of the XXZ spin-1/2 chain. The analysis yields the edge exponents and associated amplitudes that describe the local behavior of the response function near a threshold. The singular behavior is derived starting from first principles considerations: the method of analysis does not rely, at any stage, on some hypothetical correspondence with a field theory or other phenomenological approaches. The analysis builds on the massless form factor expansion for the response functions of the XXZ chain obtained recently by the author. It confirms the non-linear Luttinger based predictions relative to the power-law behavior and of the associated edge exponents that arise in the vicinity of the dispersion relation of one massive excitation (hole, particle, or bound state). In addition, the present analysis shows that due to the lack of strict convexity of the particle dispersion relation and due to the presence of slow velocity branches of the bound states, there exist excitation thresholds with a different structure of edge exponents. These originate from multi-particle/hole/bound state excitations maximizing the energy at fixed momentum.

arXiv:1811.06076•Journal of Mathematical Physics•Jun 1, 2021•12 citations

Threshold singularities in the one-dimensional Hubbard model

Fabian H. L. Eßler

We consider excitations with the quantum numbers of a hole in the one-dimensional Hubbard model below half-filling. We calculate the finite-size corrections to the energy. The results are then used to determine threshold singularities in the single-particle Green's function for commensurate fillings. We present the analogous results for the Yang-Gaudin model (electron gas with $\ensuremath{\delta}$-function interactions).

arXiv:1002.1671•Physical Review B•May 25, 2010•57 citations
Abstract We directly calculate the interaction energy of an eigenstate with a single particle and a single hole for the XXZ -spin chain. We find the connection of this addition to energy with a two-particle forward scattering phase shift function. The results can be applied to the calculation of the exponents of the threshold singularities in the correlators of the XXZ -spin chain.
Exact formulas for the singularities of the dynamical structure factor, Szz(q,omega), of the S=1/2 xxz spin chain at all q and any anisotropy and magnetic field in the critical regime are derived, expressing the exponents in terms of the phase shifts which are known exactly from the Bethe ansatz solution. We also study the long-time asymptotics of the self-correlation function 0|Sjz(t)Sjz(0)|0. Utilizing these results to supplement very accurate time-dependent density matrix renormalization group, for short to moderate times, we calculate Szz(q,omega) to very high precision.
The energy shift due to the interaction of two particles in a large box is proportional to the free particle scattering phase-shift. This provides an approximation to the effective interaction referred to as the phase-shift approximation. For a many-body fermion system this effective interaction has to be corrected for the Pauli-blocking. It is used to calculate the energy of a spin 1/2 fermion system as a function of the two-body scattering length and effective range. In the unitary limit with scattering length going to infinity and effective range going to zero the energy is 0.540 (in the units of the non-interactiong fermi gas energy) with a pp-ladder summation. Including hh-ladders the energy is 0.570. A smooth crossover from the BCS to the BEC region is observed. Pairing is not included.
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.
This work extracts, by means of an exact analysis, the singular behavior of the dynamical response functions—the Fourier transforms of dynamical two-point functions—in the vicinity of the various excitation thresholds in the massless regime of the XXZ spin-1/2 chain. The analysis yields the edge exponents and associated amplitudes that describe the local behavior of the response function near a threshold. The singular behavior is derived starting from first principles considerations: the method of analysis does not rely, at any stage, on some hypothetical correspondence with a field theory or other phenomenological approaches. The analysis builds on the massless form factor expansion for the response functions of the XXZ chain obtained recently by the author. It confirms the non-linear Luttinger based predictions relative to the power-law behavior and of the associated edge exponents that arise in the vicinity of the dispersion relation of one massive excitation (hole, particle, or bound state). In addition, the present analysis shows that due to the lack of strict convexity of the particle dispersion relation and due to the presence of slow velocity branches of the bound states, there exist excitation thresholds with a different structure of edge exponents. These originate from multi-particle/hole/bound state excitations maximizing the energy at fixed momentum.