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Physical review. B, Condensed matter

Exact representation of the large-UHubbard model and its application

Y.R. Wang

Published Jan 1, 1995• 7 citations
Publisher

Abstract

An exact representation of the large-U Hubbard model is developed in terms of fermionic holon, fermionic doublon, and local spin operators which requires no constraint between the charge and spin operators. The Hamiltonian in the new representation explicitly expresses the Nagaoka theorem, and there is an exact one to one correspondence between the new representation and the large-U Lieb-Wu wave function for the case of one dimension. Using the new representation, we further show that the signature of spin-charge separation, in any dimension, is the characteristic wave vector 2${\mathbf{k}}_{\mathit{F}}^{\mathrm{SF}}$ in the static density-density correlation function (${\mathbf{k}}_{\mathit{F}}^{\mathrm{SF}}$ encloses a Fermi volume filled by 1-x spinless fermions per site, where x is the doping concentration). In addition, we derive the corresponding representation for the t-J-J' model, where the J' term describes the indirect hopping of the electrons, and the holon (doublon) propagator which explicitly shows the dependence of charge propagation on the spin state.

DOI
10.1103/physrevb.51.234

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Physical review. B, Condensed matter•Feb 1, 1990•654 citations
An exact representation of the large-U Hubbard model is developed in terms of fermionic holon, fermionic doublon, and local spin operators which requires no constraint between the charge and spin operators. The Hamiltonian in the new representation explicitly expresses the Nagaoka theorem, and there is an exact one to one correspondence between the new representation and the large-U Lieb-Wu wave function for the case of one dimension. Using the new representation, we further show that the signature of spin-charge separation, in any dimension, is the characteristic wave vector 2kFSF{\mathbf{k}}_{\mathit{F}}^{\mathrm{SF}}kFSF​ in the static density-density correlation function (kFSF{\mathbf{k}}_{\mathit{F}}^{\mathrm{SF}}kFSF​ encloses a Fermi volume filled by 1-x spinless fermions per site, where x is the doping concentration). In addition, we derive the corresponding representation for the t-J-J' model, where the J' term describes the indirect hopping of the electrons, and the holon (doublon) propagator which explicitly shows the dependence of charge propagation on the spin state.
We propose a new fermion-spin transformation to implement the charge-spin separation in the large U Hubbard, or the equivalent t-J model. The charge degree of freedom is represented by a spinless fermion while the spin degree of freedom is represented by a hard-core boson. The local constraint for single occupancy is exactly satisfied. Very good agreement with exact solution is obtained for one-dimensional case in the mean field approximation, regarding the total energy, gapless spinon and holon spectra, and the momentum distribution of physical electrons. The same approximation yields good doping dependence of the staggered magnetization in the two-dimensional case.
A path-integral formalism for the one-dimensional Hubbard model in the strong-coupling regime, which is equivalent to the t-J model in t/U expansion but without any explicit constraint, is developed. Based on this formalism, the zero-temperature properties of the Hubbard chain are systematically studied. In the infinite-U limit, the charge and spin degrees of freedom are shown to be completely separated. Such a separation at U=\ensuremath{\infty} provides a controllable perturbative scheme to study the strong-coupling case. In the large-U regime, the spin degree of freedom is represented by a ``squeezed'' Heisenberg chain. We solve the (squeezed) Heisenberg chain by introducing a soliton representation. Both the charge and spin excitations are found to agree well with the exact solution. The bare electron (hole) is shown to be a composite particle of these basic excitations, i.e., holon and spinon, together with a nonlocal string field. This string field makes the electron behave like a ``semion'' and plays an important role in determining various correlation functions. We analytically derive the asymptotic forms of the spin-spin and density-density correlation functions as well as the single-electron and the pair-electron Green's functions. The implications of the present work to the two-dimensional model are discussed.
We have introduced a new perturbative approach for $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model where Hubbard operators are treated as fundamental objects. Using our vertices and propagators we have developed a controllable large-NNN expansion to calculate different correlation functions. We have investigated charge density-density response and the phase diagram of the model. The charge correlations functions are not very sensitive to the value of JJJ and they show collective peaks (or zero sound) which are more pronounced when they are well separated (in energy) from the particle-hole continuum. For a given JJJ a Fermi liquid state is found to be stable for doping $\ensuremath{\delta}$ larger than a critical doping ${\ensuremath{\delta}}_{c}$. ${\ensuremath{\delta}}_{c}$ decreases with decreasing JJJ. For the physical region of the parameters and, for $\ensuremath{\delta}<{\ensuremath{\delta}}_{c}$, the system enters in an incommensurate flux or DDW phase. The inclusion of the nearest-neighbor Coulomb repulsion VVV leads to a charge density wave phase when VVV is larger than a critical value Vc{V}_{c}Vc​. The dependence of Vc{V}_{c}Vc​ with $\ensuremath{\delta}$ and JJJ is shown. We have compared the results with other ones in the literature.
The one-dimensional Hubbard model is studied for general electron densities and large U. On the basis of the coupled integral equations of Lieb and Wu exact asymptotic expansions are derived to fifth order in ${U}^{\mathrm{\ensuremath{-}}1}$ for the ground-state energy, the kinetic energy, and the number of doubly occupied sites. A closed-form expression for the distribution functions of charge and spin degrees of freedom is proposed which yields the same large-U expansion as that derived directly from the Lieb-Wu equations. Our analytical results are in excellent agreement with previous numerical studies.
A new representation for electrons is introduced, in which the electron operators are written in terms of a spinless fermion and the Pauli operators. This representation is canonical, invertible, and constraint-free. Importantly, it simplifies the Hubbard interaction. On a bipartite lattice, the Hubbard model is reduced to a form in which the exchange interaction simply emerges by decoupling the Pauli subsystem from the spinless fermion bath. This exchange correctly reproduces the large UUU superexchange. Also derived, for $U=\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$, is the Hamiltonian to study Nagaoka ferromagnetism. In this representation, the infinite-UUU Hubbard problem becomes elegant and easier to handle. Interestingly, the ferromagnetism in the Hubbard model is found to be related to the gauge invariance of the spinless fermions. A generalization of this representation for the multicomponent fermions, a new representation for bosons, the notion of a ``soft-core'' fermion, and some interesting unitary transformations are introduced and discussed in the appendixes.
The momentum distribution function n(k) and spin-correlation function S(k) are determined for the one-dimensional large-U Hubbard model with various electron densities. The present work is featured with an application of the Bethe-ansatz wave function, which has a simple form in the large-U limit for any electron density. Namely, its charge degrees of freedom are expressed as a Slater determinant of spinless fermions, while its spin degrees of freedom are equivalent to the one-dimensional S=(1/2 Heisenberg model. The singularity of n(k) at k=kF{k}_{F}kF​ is analyzed from the system size dependence. In addition to the kF{k}_{F}kF​ singularity, n(k) has a weak singularity at k=3kF{k}_{F}kF​; however, no detectable singularity is present at 2kF{k}_{F}kF​, which one might expect from the spinless fermion wave function. The singularity of S(k) at 2kF{k}_{F}kF​ is also examined in detail. It is concluded from the size dependence that, when the system is away from half-filling, the nature of the singularity at 2kF{k}_{F}kF​ is different from that in the Heisenberg model. The results are compared with the behavior in the weak-correlation regime examined with the perturbation calculation, the prediction of g-ology, and recent Monte Carlo calculations.