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Temperature correlators in the one-dimensional Hubbard model in the strong coupling limit

A. G. Izergin, A. G. Pronko, N. I. Abarenkova

Published Jan 1, 1998• 22 citations
OpenAlex

Abstract

We consider the one-dimensional Hubbard model with the infinitely strong repulsion. The two-point dynamical temperature correlation functions are calculated. They are represented as Fredholm determinants of linear integrable integral operators. 1 Introduction The one-dimensional Hubbard model [1] is one of the most interesting and important model of strongly correlated electrons (see, e.g., book [2]). The hamiltonian of the model with the chemical potential h in the external constant field B describes interacting fermions on the one-dimensional periodic lattice of length L: HU = \\Gamma L X x=1 X ff=";# i c y x;ff c x\\Gamma1;ff + c y x\\Gamma1;ff c x;ff j +h L X x=1 (n x;" + n x;# ) \\Gamma B L X x=1 (n x;" \\Gamma n x;# ) + U L X x=1 n x;" n x;# : (1) Here c x;ff and c y x;ff (x = 1; . . . ; L; ff ="; #; c 0;ff j c L;ff ; c y 0;ff j c y L;ff ) are canonical fermion fields with the anticommutation relations [c x;ff ; c y y;fi ] + = ffi x;y ffi ff;fi ...

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Introduction Impenetrable bosons in one space dimension are a special case (at coupling constant c → ∞) of the one-dimensional Bose gas (NS model) considered in detail in Chapter I. The results obtained there for the NS model can be specialized to the case of impenetrable bosons. It is to be said that all the formulæe are considerably simplified in this case. From the point of view of physics this is due to the fact that the δ-function potentials in the N -particle Hamiltonian (I.1.11) are now infinitely strong and the bosons cannot penetrate one another, so that the wave function should be equal to zero if the bosons' coordinates coincide. In this respect impenetrable bosons are rather similar to free fermions (see Appendix I.I). In this chapter correlation functions of impenetrable bosons are considered and their representations as Fredholm determinants of linear integral operators are given. The (very essential) simplification with respect to the general case is due to the fact that now all the auxiliary quantum fields can be set equal to zero (see IX.6), so that the kernels of these operators do not contain quantum fields. The determinant representation for the time-dependent correlator is easy to obtain in this case. Temperature correlation functions are considered as well as zero temperature ones.
The authors show with the aid of a canonical transformation that separates charge and spin degrees of freedom that the strongly correlated one-dimensional Hubbard model with infinite (U= infinity ) on-site repulsion is related to a corresponding spinless free-fermion lattice model. Thermodynamic properties of the two models are shown to be closely related whereas the single-particle two-site function for the U= infinity model is shown to be related to a modified many site function for the spinless model which involves many-point correlations. The latter is shown to be expressible in terms of the inverse of a certain matrix which should be amenable to numerical and analytical analysis.
Analytical results on the correlation functions of strongly correlated many-body systems are rare in the literature and their importance cannot be overstated. We present determinant representations for the space-, time-, and temperature-dependent correlation functions of the strongly interacting one-dimensional Hubbard model in the presence of an external trapping potential. These representations are exact and valid in both equilibrium and nonequilibrium scenarios like the ones initiated by a sudden change of the confinement potential. In addition, they can be implemented numerically very easily, significantly outperforming other numerical approaches. As applications of our results we investigate the single-particle spectral functions of systems with harmonic trapping and show that dynamical quasicondensation occurs for both fermionic and bosonic spin-1/21/21/2 systems released from a Mott insulator state.
Exact relations are derived for the Fermi Hubbard spectral weight function for infinite U at zero temperature in the thermodynamic limit for any dimension,any lattice structure and general hopping matrix. These relations involve moments of the spectral weight function but differ from similar work by (a) restricting the moments over the interesting low energy (lower Hubbard band) spectrum and (b) without any of the usual approximations (e.g. decoupling) for the requisite higher order correlation functions. The latter are determined exactly as explicit functions of the single particle correlation function in the infinite U limit. The methods behind these results have the additional virtue of simplicity - cumbersome, but entirely straightforward. In a companion paper, the relations offer rich insights into the model behavior and offer severe tests - a computer lab - of any proposed solution to the Hubbard model in the strong correlation (large finite U) region. As the electron density approaches one electron per site, then the correlation functions become local, so are trivial to compute exactly. In that limit and for the paramagnetic case, the second central moment of the spectral weight function is not zero, but quite large.