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Physical review. B, Condensed matter

Bosonization of the constrained spinon-holon model

D. Schmeltzer

Published Mar 1, 1994• 6 citations
Publisher

Abstract

We present a bosonization method for one-dimensional nonrelativistic fermions with spin. This method is based on the Jordan-Wigner transformation and can be used directly in the strong-coupling limit. We apply this method to the constrained spinon-holon model (at each site we have either a holon or a spinon, in each bond the holon and spinon current vanishes) and show that this model is a Luttinger liquid.

DOI
10.1103/physrevb.49.6944

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Physical review. B, Condensed matter•May 1, 1993•14 citations
We present a bosonization method for one-dimensional nonrelativistic fermions with spin. This method is based on the Jordan-Wigner transformation and can be used directly in the strong-coupling limit. We apply this method to the constrained spinon-holon model (at each site we have either a holon or a spinon, in each bond the holon and spinon current vanishes) and show that this model is a Luttinger liquid.
The bosonization scheme for constrained fermions in one dimension (1D) is constructed. It is shown that the constrained bosonization is similar to the unconstrained one: the only difference is a phase shift simple. This bosonization scheme is applied to the 1D t-J model. For small J/t the model is equivalent to the Luttinger liquid for charge and spin. When J/t\ensuremath{\ge}\ensuremath{\pi} sin(\ensuremath{\pi}\ensuremath{\delta})/cos(2\ensuremath{\pi}\ensuremath{\delta}) (with doping \ensuremath{\delta}) the holon liquid is unstable and one has only a spin liquid. This corresponds to phase separation.
The Jordan-Wigner transformation is traditionally applied to one dimensional systems, but recent works have generalized the transformation to fermionic lattice systems in higher dimensions while keeping locality manifest. These developments could aid the theoretical or even experimental studies of strongly correlated electronic problems through their bosonic counterparts. In this work, we develop a scheme for higher-dimensional Jordan-Wigner transformation which keeps all relevant symmetries manifest on the bosonic side. Our approach connects the discussion of exact lattice bosonization to the familiar notions of fractionalized partons like spinons and chargeons, and works for spin-1/21/21/2 fermions -- like the physical electrons -- on four-coordinated lattices. The construction is applied to fermions defined on the square, kagome and diamond lattices, and we provide explicit expressions for the bosonized versions of well-known models of strongly correlated electrons, like the Hubbard and ttt-JJJ models.
The Hubbard model in the strong coupling limit is considered in two dimensions. A bosonization method based on the extention of the Jordan-Wigner transformation to particles with spin and no double occupancy is introduced. We find that at T=0 a Fermi liquid exists and superconductivity appears. For a finite temperature and finite doping a normal state populated with vortices exists. As a result, we have a normal state with separated charge- and spin-density-wave excitations. Formally those excitations can be described with the help of the Luttinger-liquid representation. In the second part we parametrize the Fermi surface by a set of N points. Around each point we describe the Fermi surface excitations by holons and spinons and find a set of N-coupled Luttinger channels. At T=0 we have space-time tunneling of electric charges which destroys the Luttinger liquid in favor of a Fermi liquid. At T\ensuremath{\ne}0 or LT\ensuremath{\rightarrow}\ensuremath{\infty} (L is the size of the system) tunneling of electric charges might be prohibited giving rise to gapless excitations. In the last part we consider the case with a commensurate frozen flux in the ground state. As a result we find that the excitations are semions.
The composite-spin representation of the spin-1 Heisenberg chain is used to transform it through the Jordan-Wigner transformation to the one-dimensional fermion gas. To properly include the xy couplings between spins, we also consider the bosonized version of the fermion model. Phase diagrams deduced from the two versions of the fermion model are compared against numerical results for finite Heisenberg chains. One of the symmetries of the spin model is lost in the fermionization, and this leads to a topologically incorrect phase diagram in at least one part of the parameter space. There are clear indications of significant coupling of spin and charge degrees of freedom in the fermion model and of strong renormalization of the coupling constants in the continuum limit. The existence of the Haldane gap is discussed in connection with these results.
Using the Jordan-Wigner transformation and duality relations, a method for bosonization for nonrelativistic spinless fermions is introduced. The same method is used for one- and two-dimensional cases. In one dimension a cubic potential is obtained. After linearization the usual bosonization results for relativistic fermions are reproduced. In two dimensions the nonlocal potential induces vortices which determines the effective potential between the charged particles. We find that the low-energy excitations are acoustic charge-density waves.