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

Chinese Physics Letters

Phase Operator and Lattice Green's Function for Ring-Lattice Model inp-th Nearest Neighbour Interaction Approximation

Hong-Yi Fan, Xiao‐Yin Pan

Published Aug 1, 1998• 2 citations
Publisher

Abstract

In dealing with the one-dimensional lattice model, we replace the traditionally needed Born-Von-Karmann periodic boundary condition with two additional Hamiltonian terms to make up a ring-lattice. In so doing, a unitary phase operator and the corresponding Hermite phase angle operator can be introduced as those in quantum optics theory. The new Hamiltonian is invariant under the phase shift transformation. Moreover, the lattice Green's function in the p th nearest interaction approximation can also be easily derived from the new Hamiltonian, which becomes the well-known lattice Green's function when the ring is infinite.

DOI
10.1088/0256-307x/15/8/001

Related papers

GREEN'S FUNCTION OF A FINITE CHAIN AND THE DISCRETE FOURIER TRANSFORM

S. Cojocaru

Unlike the Fourier series expansion, the discrete Fourier transform is defined on a finite basis set of harmonic functions. The first approach is widely used in condensed matter to describe the thermodynamic limit of various lattice models, while the latter did not receive sufficient development that would allow to address finite lattices. In the present paper a general expression for the Green's function of a finite one-dimensional lattice with nearest neighbor interaction is derived for the first time via discrete Fourier transform. Solution of the Heisenberg spin chain with periodic and open boundary conditions is considered as an example. Although the final expressions are completely equivalent to Bethe ansatz, the examples allow us to clarify the differences between the two approaches. On the other hand, it is explained why the well known results obtained by Fourier series expansion were incomplete and thus provides a deeper understanding of the approach.

arXiv:0704.2898•International Journal of Modern Physics B•Feb 20, 2006•10 citations

Green's function and excitation spectrum of finite lattices

S. Cojocaru, Victor Bârsan, Arnout Jozef Ceulemans

Abstract New analytical results are obtained for the Green's function of a finite one‐dimensional lattice with nearest neighbor interaction using discrete Fourier transform. The method offers an alternative to the Bethe Ansatz, but does not require any a priori assumption on the form of the wavefunction. This makes it suitable for extensions to nano‐ferromagnets of higher dimensions. Solutions of the Heisenberg spin chain with periodic and open boundary conditions are considered as examples. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

physica status solidi (b)•Jun 1, 2006•1 citation

Few-Particle Green’s Functions for Strongly Correlated Systems on Infinite Lattices

Mona Berciu

We show how few-particle Green's functions can be calculated efficiently for models with nearest-neighbor hopping, for infinite lattices in any dimension. As an example, for one-dimensional spinless fermions with both nearest-neighbor and second-nearest-neighbor interactions, we investigate the ground states for up to 5 fermions. This allows us not only to find the stability region of various bound complexes, but also to infer the phase diagram at small but finite concentrations.

arXiv:1112.1928•Physical Review Letters•Dec 7, 2011•26 citations

Unitary Operator and Density Matrix for a Ring of Coupled Oscillator Model

Fan Hong‐Yi, Xu Zhihua, Fan Yue

A unitary operator which can directly diagonalize the Hamiltonian of a ring of coupled oscillators is found. With use of the technique of integration within an ordered product of operators we see that includes a squeezing transformation. For large it turns out that, except a zero-mode, the ring Hamiltonian has the same spectrum as the one-dimensional monatomic lattice model plus the Born–Von–Karmen boundary condition. The density matrix of this model is calculated by the transformation, which further leads us to derive the heat capacity of the coupled oscillators in the large limit.

Communications in Theoretical Physics•Oct 30, 1998•2 citations

Finite lattice methods in one and two dimensional quantum spin systems

Phillip M. Duxbury

This is a study of finite lattice methods applied to quantum spin models on one and two dimensional lattices.The motivation for the use of these techniques is provided by placing them in the context of the wider field of approximation methods in the statistical mechanics of spin models.After reviewing previous finite lattice calculations on quantum spin models, the theory needed to efficiently generate and analyse finite lattice data is given.Finite lattice calculations on three quantum spin models are then presented.The first calculation provides an estimate of the transverse susceptibility of the one dimensional antiferromagnetic XY model, by extrapolation from calculations on rings of up to 10 sites.Good convergence is found for KB T ~ 0•4 and these results are compared with recent c -axis susceptibility measurements on the compound Cs 2 CoC1 4 .Chapters 5 and 6 study the quantum hamiltonian analog of the two dimensional axial next nearest neighbour Ising model, A perturbation analysis using non degenerate Rayleigh-Schrodingerperturbation theory is followed by a mass gap scaling analysis.These analyses provide accurate phase boundaries for, and ellucidate several interesting features of, the phase diagram of this model.In particular, the model is shown to exhibit a phase whose modulation varies continuously with a change in the interaction parameters in the hamiltonian.This "modulated" phase region lies between a normal ordered phase and a paramagnetic phase.Evid~nce is given that the transition from the paramagnetic to modulated phase is exponential in nature.The third model studied is a spin 1 biquadratic Heisenberg model with hamiltonian,<ij> (Jl s .. s. +J2(~ . .~.)2). 1 J 1 J Accurate estimates of the ground state energy in both one and two dimensions are found by calculating this quantity on 4,6,8 and 10 site lattices.Comment is made about the ground state phase diagram, and about the existence of intermediate spin states and states with a complicated magnetic ordering.The thesis concludes by commenting on the performance of finite lattice methods in the three models studied and by outlining certain directions for future advance., ACKNOWLEVGEMENTS I would like to express

•Mar 18, 2022•0 citations

Hubbard model in one dimension with a general bond-charge interaction: Analytical ground-state solution for the pairing of two particles

Emmanuel Vallejo, Oracio Navarro

In this work we solve a general Hubbard Hamiltonian for two interacting particles in a periodic and nonperiodic infinite one-dimensional lattice, using a real-space mapping method, the renormalized perturbation expansion (RPE), and the Green-function technique. This Hamiltonian considers a general bond-charge interaction, the on-site interaction, and the general intersite interaction. The real-space method is based on mapping the correlated many-body problem onto an equivalent site- and bond-impurity tight-binding problem in a higher-dimensional space. Analyzing the periodic and the quasiperiodic lattices in this new space, we obtained the analytical solution for the binding condition at the ground state. Our general results for the periodic chain reproduce completely the limit cases of the numerical solution obtained previously and those obtained in reciprocal space

Physical review. B, Condensed matter•May 28, 2003•5 citations
In dealing with the one-dimensional lattice model, we replace the traditionally needed Born-Von-Karmann periodic boundary condition with two additional Hamiltonian terms to make up a ring-lattice. In so doing, a unitary phase operator and the corresponding Hermite phase angle operator can be introduced as those in quantum optics theory. The new Hamiltonian is invariant under the phase shift transformation. Moreover, the lattice Green's function in the p th nearest interaction approximation can also be easily derived from the new Hamiltonian, which becomes the well-known lattice Green's function when the ring is infinite.
Unlike the Fourier series expansion, the discrete Fourier transform is defined on a finite basis set of harmonic functions. The first approach is widely used in condensed matter to describe the thermodynamic limit of various lattice models, while the latter did not receive sufficient development that would allow to address finite lattices. In the present paper a general expression for the Green's function of a finite one-dimensional lattice with nearest neighbor interaction is derived for the first time via discrete Fourier transform. Solution of the Heisenberg spin chain with periodic and open boundary conditions is considered as an example. Although the final expressions are completely equivalent to Bethe ansatz, the examples allow us to clarify the differences between the two approaches. On the other hand, it is explained why the well known results obtained by Fourier series expansion were incomplete and thus provides a deeper understanding of the approach.
Abstract New analytical results are obtained for the Green's function of a finite one‐dimensional lattice with nearest neighbor interaction using discrete Fourier transform. The method offers an alternative to the Bethe Ansatz, but does not require any a priori assumption on the form of the wavefunction. This makes it suitable for extensions to nano‐ferromagnets of higher dimensions. Solutions of the Heisenberg spin chain with periodic and open boundary conditions are considered as examples. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
A unitary operator which can directly diagonalize the Hamiltonian of a ring of coupled oscillators is found. With use of the technique of integration within an ordered product of operators we see that includes a squeezing transformation. For large it turns out that, except a zero-mode, the ring Hamiltonian has the same spectrum as the one-dimensional monatomic lattice model plus the Born–Von–Karmen boundary condition. The density matrix of this model is calculated by the transformation, which further leads us to derive the heat capacity of the coupled oscillators in the large limit.
This is a study of finite lattice methods applied to quantum spin models on one and two dimensional lattices.The motivation for the use of these techniques is provided by placing them in the context of the wider field of approximation methods in the statistical mechanics of spin models.After reviewing previous finite lattice calculations on quantum spin models, the theory needed to efficiently generate and analyse finite lattice data is given.Finite lattice calculations on three quantum spin models are then presented.The first calculation provides an estimate of the transverse susceptibility of the one dimensional antiferromagnetic XY model, by extrapolation from calculations on rings of up to 10 sites.Good convergence is found for KB T ~ 0•4 and these results are compared with recent c -axis susceptibility measurements on the compound Cs 2 CoC1 4 .Chapters 5 and 6 study the quantum hamiltonian analog of the two dimensional axial next nearest neighbour Ising model, A perturbation analysis using non degenerate Rayleigh-Schrodingerperturbation theory is followed by a mass gap scaling analysis.These analyses provide accurate phase boundaries for, and ellucidate several interesting features of, the phase diagram of this model.In particular, the model is shown to exhibit a phase whose modulation varies continuously with a change in the interaction parameters in the hamiltonian.This "modulated" phase region lies between a normal ordered phase and a paramagnetic phase.Evid~nce is given that the transition from the paramagnetic to modulated phase is exponential in nature.The third model studied is a spin 1 biquadratic Heisenberg model with hamiltonian,<ij> (Jl s .. s. +J2(~ . .~.)2). 1 J 1 J Accurate estimates of the ground state energy in both one and two dimensions are found by calculating this quantity on 4,6,8 and 10 site lattices.Comment is made about the ground state phase diagram, and about the existence of intermediate spin states and states with a complicated magnetic ordering.The thesis concludes by commenting on the performance of finite lattice methods in the three models studied and by outlining certain directions for future advance., ACKNOWLEVGEMENTS I would like to express
In this work we solve a general Hubbard Hamiltonian for two interacting particles in a periodic and nonperiodic infinite one-dimensional lattice, using a real-space mapping method, the renormalized perturbation expansion (RPE), and the Green-function technique. This Hamiltonian considers a general bond-charge interaction, the on-site interaction, and the general intersite interaction. The real-space method is based on mapping the correlated many-body problem onto an equivalent site- and bond-impurity tight-binding problem in a higher-dimensional space. Analyzing the periodic and the quasiperiodic lattices in this new space, we obtained the analytical solution for the binding condition at the ground state. Our general results for the periodic chain reproduce completely the limit cases of the numerical solution obtained previously and those obtained in reciprocal space