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arXiv:0907.2740

Green functions and correlation functions of a solvable S=1 quantum Ising spin model with dimerization

Yang, Zhi-Hua, Li‐Ping Yang, Haina Wu, Jianhui Dai, Tao Xiang

Published Jul 16, 2009• 0 citations
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Abstract

This is a supplementary material of our recent paper\cite{yangPRB}, where a class of exactly solvable S=1 quantum Ising spin models were studied based on the hole decomposition scheme. Here we provide some details for the Green functions, the spin-spin correlation functions, as well as the spin susceptibility in the presence of dimerization.

DOI
10.48550/arxiv.0907.2740

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Димер в модели Хаббарда

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Background. Exactly solvable models are of great importance in quantum physics, as most models in quantum physics are not exactly solvable. The detailed analysis of exactly solvable model’s behavior can be made by exact solutions. The solutions of exactly solvable models, obtained by the approximation method, can be compared with exact solutions, used as standards. A dimer in the Hubbard model is an exactly solvable model. The purpose of this paper is to compare exact solutions obtained for a dimer in the Hubbard model, with solutions obtained for a dimer by approximation of static fluctuations. Materials and methods. In this study, to construct exact solutions and solutions in approximation of static fluctuations the authors used the method of motion equations for creation operators, the method of Green’s functions and the spectral theorem. The energy spectrum of a quantum system can be calculated by Green’s functions, as the poles of Green’s functions correspond to the energy spectrum of a quantum system. Knowing the Green’s functions, the spectral theorem allows to calculate the correlation functions of the quantum system. Results. Resulting from the usage of the methods of the quantum field theory there were obtained anticommutator Green’s functions for a dimer, that were both exact and in approximation of static fluctuations. Using the anticommutator Green’s functions the authors obtained correlation functions and the energy spectrum for a dimer. Conclusions. This work demonstrates that the energy spectrum of a dimer is split by a strong Coulomb repulsion of electrons on two Hubbard subbands and the “metal-insulator” transition occurs in a dimer. This work also demonstrates that several energy states of a dimer, obtained by the approximation of static fluctuations, coincide with energy states of a dimer, obtained by an exact solution.

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This is a supplementary material of our recent paper\cite{yangPRB}, where a class of exactly solvable S=1 quantum Ising spin models were studied based on the hole decomposition scheme. Here we provide some details for the Green functions, the spin-spin correlation functions, as well as the spin susceptibility in the presence of dimerization.
The one-dimensional spin-1/2 Ising-Heisenberg ferromagnets are studied by the Green's function method at finite temperatures. The Green's function is decoupled in terms of the correlation functions of 〈S0z{\mathit{S}}_{0}^{\mathit{z}}S0z​Snz{\mathit{S}}_{\mathit{n}}^{\mathit{z}}Snz​〉 and 〈S0+{\mathit{S}}_{0}^{+}S0+​${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈Sz{\mathit{S}}^{\mathit{z}}Sz〉=0, from which a set of self-consistent equations are derived and solved numerically. The thermodynamic properties such as internal energy, specific heat, and zero-field parallel susceptibility are calculated. Our results predict the correct behavior for the correlation functions and thermodynamic quantities in the entire temperature region, in agreement both with the exact numerical results on the finite chain and the experimental data.
We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for T>TcT>{T}_{c}T>Tc​ and T<TcT<{T}_{c}T<Tc​. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where ttt is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-ttt behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.
We calculate the spin-spin correlations and the magnetic susceptibility of the quantum one- and two-dimensional XY\mathrm{XY}XY models with S=1,S=1,S=1, using the two-time Green's function method and performing a decoupling proposed by Kondo and Yamaji for cases with no long-range order. A set of self-consistent equations of the correlation functions are derived and solved numerically. We present results for the correlation functions, susceptibilities, and specific heat for the whole range of temperature.
For both T>TcT>{T}_{c}T>Tc​ and T<TcT<{T}_{c}T<Tc​, we report exact expressions for the nnn-spin correlation function of the two-dimensional Ising model suitable for studying large separations between spins. In particular, the scaling limit of these correlation functions can be shown to exist and yields nnn-point Schwinger functions of some relativistic quantum field theories.
Background. Exactly solvable models are of great importance in quantum physics, as most models in quantum physics are not exactly solvable. The detailed analysis of exactly solvable model’s behavior can be made by exact solutions. The solutions of exactly solvable models, obtained by the approximation method, can be compared with exact solutions, used as standards. A dimer in the Hubbard model is an exactly solvable model. The purpose of this paper is to compare exact solutions obtained for a dimer in the Hubbard model, with solutions obtained for a dimer by approximation of static fluctuations. Materials and methods. In this study, to construct exact solutions and solutions in approximation of static fluctuations the authors used the method of motion equations for creation operators, the method of Green’s functions and the spectral theorem. The energy spectrum of a quantum system can be calculated by Green’s functions, as the poles of Green’s functions correspond to the energy spectrum of a quantum system. Knowing the Green’s functions, the spectral theorem allows to calculate the correlation functions of the quantum system. Results. Resulting from the usage of the methods of the quantum field theory there were obtained anticommutator Green’s functions for a dimer, that were both exact and in approximation of static fluctuations. Using the anticommutator Green’s functions the authors obtained correlation functions and the energy spectrum for a dimer. Conclusions. This work demonstrates that the energy spectrum of a dimer is split by a strong Coulomb repulsion of electrons on two Hubbard subbands and the “metal-insulator” transition occurs in a dimer. This work also demonstrates that several energy states of a dimer, obtained by the approximation of static fluctuations, coincide with energy states of a dimer, obtained by an exact solution.