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Physics of the Solid State

Two-particle states in the 2D Hubbard model

A. S. Saakyan

Published May 1, 2000• 0 citations
Publisher

Abstract

An exact solution is proposed for the problem of two singlet electrons (zero-spin bosons) interacting through a Hubbard-type potential on a bounded quadratic lattice. Exact two-particle states and the energy spectrum are constructed.
DOI
10.1134/1.1131319

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SZFKI Annual Report 1995

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The Hubbard model for the square planar lattice in the limit t/U<<1 is discussed. The ground state is chosen to be the product of the Hubbard model's exact solutions for four-site blocks. One-particle excitations appear to be Fermi particles with spin 1 / 2 . Their effective mass near the bottom of the band coincides with the effective mass of excitations in the paramagnetic state in the 'Hubbard-I' approximation. However, minima of the spectrum are situated at other points in the Brillouin zone. The possibility of the binding of two excitations into singlet pair with charge 2e is demonstrated. Cohesive energy is calculated to be equal to 0.1t. The superfluid state of the Bose gas of such pairs is assumed to be connected with high-T c superconductivity.
Low dimensional magnetic models.-The one-dimensional Heisenberg model is known to be soluble by the Bethe Ansatz.The spectrum of a ferromagnet consists of magnons and bound state of magnons.The problem of finding the bound state of a macroscopically large number of magnons poses a special problem.These states are described by macroscopically long string solutions of the Bethe Ansatz equations.To find such solutions numerically we developed a program by which even chains of several hundred sites can be treated.Our results show that, as expected, the structure of the macroscopically long strings is not of the Bethe-type.Recently several new materials have been discovered in which the localized magnetic moments are situated in a chain like structure.The energy spectrum and the behaviour of these systems depends strongly on the value of the spin and the number of coupled chains.We started to study these models using the density matrix renormalization group.One-dimensional fermionic models.-The Hubbard model is the simplest model of the interacting electron system.It is usually formulated on a discrete lattice.Sometimes it is more appropriate to work in the continuum limit.We have shown, that the continuum limit of the Hubbard chain, in which the particle number per site is kept constant leads to a relativistic model describing both finite and zero mass particles.By constructing the relativistic limit of the half-filled band we have shown, that also the relativistic particles are described by Bethe-Ansatz type equations, while in the relativistic limit of the non-half-filled chain this is true only for the massive particles, and the equations of the massless particles are of somewhat different structure.Theory of dissipative motion of heavy particles.-We studied the low energy fixed point structure of an N f -flavor two level system (TLS) model, where the spin of the conduction electrons is replaced by a flavor quantum number, f= 1, ... , N f .Using the multiplicative renormalization group we constructed the (1/N f ) 2 order scaling equations and we analyzed their fixed point structure.We showed that the fixed point structure of this Hamiltonian is equivalent to that of the N f -channel Kondo model and we found that the exponent of the leading irrelevant operators is also the same.However, for a TLS an infinite number of leading irrelevant operators appear which give a contribution to the different physical quantities but do not change their critical exponents.