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WORLD SCIENTIFIC eBooks

Models with Inverse-Square Exchange

Published Sep 1, 1996• 0 citations
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Abstract

A one-dimensional quantum N-body system of either fermions or bosons with SU (n) "spins" (or colors in particle physics language) interacting via inverse-square exchange is presented in this article.A class of eigenstates of both the continuum and lattice version of the model Hamiltonians is constructed in terms of the Jastrow-product type wave function.The class of states we construct in this paper corresponds to the ground state and the low energy excitations of the model that can be described by the effective harmonic fluid Hamiltonian.By expanding the energy about the ground state we find the harmonic fluid parameters (i.e. the charge, spin velocities, etc.), explicitly.The correlation exponent and the compressibility are also found.As expected the general harmonic relation(i.e.v S = (v N v J ) 1/2 ) is satisfied among the charge and spin velocities.

DOI
10.1142/9789812830951_0004

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A one-dimensional quantum N-body system of either fermions or bosons with SU (n) "spins" (or colors in particle physics language) interacting via inverse-square exchange is presented in this article.A class of eigenstates of both the continuum and lattice version of the model Hamiltonians is constructed in terms of the Jastrow-product type wave function.The class of states we construct in this paper corresponds to the ground state and the low energy excitations of the model that can be described by the effective harmonic fluid Hamiltonian.By expanding the energy about the ground state we find the harmonic fluid parameters (i.e. the charge, spin velocities, etc.), explicitly.The correlation exponent and the compressibility are also found.As expected the general harmonic relation(i.e.v S = (v N v J ) 1/2 ) is satisfied among the charge and spin velocities.
A one-dimensional quantum N-body system of either fermions or bosons with SU(n) ``spins'' (or colors in particle physics language) interacting via inverse-square exchange is presented in this paper. A class of eigenstates of both the continuum and lattice version of the model Hamiltonians is constructed in terms of the Jastrow-product-type wave function. The class of states we construct in this paper corresponds to the ground state and the low-energy excitations of the model that can be described by the effective harmonic fluid Hamiltonian. By expanding the energy about the ground state we find the harmonic fluid parameters (i.e., the charge, spin velocities, etc.) explicitly. The correlation exponent and the compressibility are also found. As expected, the general harmonic relation [i.e., vS{\mathit{v}}_{\mathit{S}}vS​=(vN{\mathit{v}}_{\mathit{N}}vN​vJ{\mathit{v}}_{\mathit{J}}vJ​)1/2{)}^{1/2})1/2] is satisfied among the charge and spin velocities.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
We identify exact excitation content of the intermediate states for the one-particle Green's functions, spin-spin, and (charge) density-density correlation functions of the periodic one-dimensional $t\ensuremath{-}J$ model with inverse-square exchange. The excitations consist of neutral S=12S=\frac{1}{2}S=21​ spinons and spinless (charge $\ensuremath{-}e$ holons with semionic fractional statistics and bosonic (charge +2e+2e+2e) "antiholons" which are excitations of the holon condensate. We find a set of selection rules and the regions of nonvanishing spectral weight in the energy-momentum space for the various correlation functions.
We construct an exactly solvable quantum impurity model which consists of spin-1/2 conduction fermions and a spin-1/2 magnetic moment. The ground state is a Gutzwiller projected Fermi sea with nonorthonormal modes and its wave function in the site-occupation basis is a Jastrow-type homogeneous polynomial. The parent Hamiltonian has all-to-all inverse-square hopping terms between the conduction fermions and inverse-square spin-exchange terms between the conduction fermions and the magnetic moment. The low-lying energy levels, spin-spin correlation function, and von Neumann entanglement entropy of our model demonstrate that it exhibits the essential aspects of spin-1/2 Kondo physics. The machinery developed in this work can generate many other exactly solvable quantum impurity models.
We present the exact solution to a one-dimensional multicomponent quantum lattice model interacting by an exchange operator which falls off as the inverse sinh square of the distance. This interaction contains a variable range as a parameter and can thus interpolate between the known solutions for the nearest-neighbor chain and the inverse-square chain. The energy, susceptibility, charge stiffness, and the dispersion relations for low-lying excitations are explicitly calculated for the absolute ground state, as a function of both the range of the interaction and the number of species of fermions.