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:cond-mat/9204017 · Physical review. B, Condensed matter

Models with inverse-square exchange

Zou Ha, F. D. M. Haldane

Published Oct 15, 1992• 163 citations
arXivHTMLPDF

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 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., ${\mathit{v}}_{\mathit{S}}$=(${\mathit{v}}_{\mathit{N}}$${\mathit{v}}_{\mathit{J}}$${)}^{1/2}$] is satisfied among the charge and spin velocities.

DOI
10.1103/physrevb.46.9359

Related papers

Models with Inverse-Square Exchange

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.

WORLD SCIENTIFIC eBooks•Sep 1, 1996•0 citations

'Luttinger liquid theory' of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas

F. D. M. Haldane

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.

Journal of Physics C Solid State Physics•Jul 10, 1981•2,495 citations

Elementary Excitations of One-Dimensional t−J Model with Inverse-Square Exchange

Zou Ha, F. D. M. Haldane

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 $S=\frac{1}{2}$ spinons and spinless (charge $\ensuremath{-}e$ holons with semionic fractional statistics and bosonic (charge $+2e$) "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.

arXiv:cond-mat/9406059•Physical Review Letters•Nov 21, 1994•38 citations

Exact quantum solutions of extraordinary N -body problems

D. Lynden-Bell, R. M. Lynden-Bell

The wave functions of boson and fermion gases are known even when the particles have harmonic interactions. Here we generalize these results by solving exactly the N–body Schrödinger equation for potentials V that can be any function of the sum of the squares of the distances of the particles from one another in three dimensions. For the harmonic case that function is linear in r2. Explicit N–body solutions are given when U(r) = –2Mh–2V(r) = 𝛇r–1 – 𝛇2r–2. Here M is the sum of the masses and r2 = 1/2 M–2ΣmImJ(xI – xJ)2. For general U(r) the solution is given in terms of the one– or two–body problem with potential U(r) in three dimensions. The degeneracies of the levels are derived for distinguishable particles, for bosons of spin zero and for spin 1/2 fermions. The latter involve significant combinatorial analysis which may have application to the shell model of atomic nuclei. For large N the fermionic ground state gives the binding energy of a degenerate white dwarf star treated as a giant atom with an N–body wave function. The N–body forces involved in these extraordinary N–body problems are not the usual sums of two–body interactions, but nor are forces between quarks or molecules. Bose–Einstein condensation of particles in three dimensions interacting via these strange potentials can be treated by this method.

arXiv:cond-mat/9904019•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Sep 8, 1999•11 citations

Exact spectrum of SU(n) spin chain with inverse square exchange

Alexios P. Polychronakos

The spectrum and partition function of a model consisting of SU(n) spins positioned at the equilibrium positions of a classical Calogero model and interacting through inverse–square exchange are derived. The energy levels are equidistant and have a high degree of degeneracy, with several SU(n) multiplets belonging to the same energy eigenspace. The partition function takes the form of a q– deformed polynomial. This leads to a description of the system in terms of an effective parafermionic hamiltonian, and to a classification of the states in terms of “modules ” consisting of base–n strings of integers.

•Jan 1, 1994•63 citations

Long-range interactions in the quantum many-body problem in one dimension: Ground state

Saugata Ghosh

We investigate the ground state properties of a family of N-body systems in one dimension, trapped in a polynomial potential and having long-range two-body interaction in addition to the inverse square potential studied in the Calogero-Sutherland model (CSM). We show that for such a Hamiltonian, the ground state energy is similar to that of free fermions in a harmonic well with a displacement that depends on the number of particles and depth of the well. We obtain the ground state wave function and using random matrix results, study the particle density and pair correlation function (PCF). We observe that the particles are arranged in bands. Due to the presence of long-range interaction, the PCF shows a departure from the CSM.

arXiv:quant-ph/0401116•Physical Review E•Mar 30, 2004•2 citations
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​=({\mathit{v}}_{\mathit{N}}$${\mathit{v}}_{\mathit{J}}$${)}^{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 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.
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.
The wave functions of boson and fermion gases are known even when the particles have harmonic interactions. Here we generalize these results by solving exactly the N–body Schrödinger equation for potentials V that can be any function of the sum of the squares of the distances of the particles from one another in three dimensions. For the harmonic case that function is linear in r2. Explicit N–body solutions are given when U(r) = –2Mh–2V(r) = 𝛇r–1 – 𝛇2r–2. Here M is the sum of the masses and r2 = 1/2 M–2ΣmImJ(xI – xJ)2. For general U(r) the solution is given in terms of the one– or two–body problem with potential U(r) in three dimensions. The degeneracies of the levels are derived for distinguishable particles, for bosons of spin zero and for spin 1/2 fermions. The latter involve significant combinatorial analysis which may have application to the shell model of atomic nuclei. For large N the fermionic ground state gives the binding energy of a degenerate white dwarf star treated as a giant atom with an N–body wave function. The N–body forces involved in these extraordinary N–body problems are not the usual sums of two–body interactions, but nor are forces between quarks or molecules. Bose–Einstein condensation of particles in three dimensions interacting via these strange potentials can be treated by this method.
The spectrum and partition function of a model consisting of SU(n) spins positioned at the equilibrium positions of a classical Calogero model and interacting through inverse–square exchange are derived. The energy levels are equidistant and have a high degree of degeneracy, with several SU(n) multiplets belonging to the same energy eigenspace. The partition function takes the form of a q– deformed polynomial. This leads to a description of the system in terms of an effective parafermionic hamiltonian, and to a classification of the states in terms of “modules ” consisting of base–n strings of integers.
We investigate the ground state properties of a family of N-body systems in one dimension, trapped in a polynomial potential and having long-range two-body interaction in addition to the inverse square potential studied in the Calogero-Sutherland model (CSM). We show that for such a Hamiltonian, the ground state energy is similar to that of free fermions in a harmonic well with a displacement that depends on the number of particles and depth of the well. We obtain the ground state wave function and using random matrix results, study the particle density and pair correlation function (PCF). We observe that the particles are arranged in bands. Due to the presence of long-range interaction, the PCF shows a departure from the CSM.