Nambu-Goldstone Modes in a Hubbard Model

Associated with breaking of continuous symmetry are low-energy excitations, known as Nambu-Goldstone modes. Crystallization comes with phonons, while ferromagnetization comes with spin waves. Pions are viewed as (pseudo-)Nambu-Goldstone bosons associated with the breaking of (approximate) chiral symmetry of quantum chromodynamics. Breakdown of gauge symmetry entails Nambu-Goldstone modes to be "eaten" by gauge fields, rendering gauge force short-ranged in the bulk of material or of universe.

Here, we shall present a simple model and a calculational method centering around this phenomenon.

A Hubbard Model and Antiferromagnetization

Let us consider the one-band Hubbard Model on a 2-dimensional square-lattice. Hamiltonian is given by

with t and U positive.

Fixing the number of electrons by the condition N_e=N_site, this model is expected to show antiferromagnetization for a macroscopic sample at low temperature, breaking its rotational symmetry (see my first report).

Spin-Spin Correlations

Once symmetry is broken, we expect low-frequency modes arising from configurations with slowly-varying order parameters. They should in turn manifest themselves as poles in various correlation functions at very low frequencies.

We will look at time-ordered spin-spin correlations in frequency space(set ħ=1):

where q ranges over half the original Brillouin zone, and Qa=(π, π). They can be measured, for example, via neutron scattering experiments.

Within Hartree-Fock

The most simple way to evaluate it is to ignore effects of interactions between two Bloch-orbitals. Feynman diagrammatically, we retain only

This diagram gives: (see my first report for the definitions of various symbols)

with ΔE_{level spacing}<<η<<(t_experiment)^-1 as usual.

Within this approximation, we cannot find poles for |ω| <E_gap, and are urged to go a little beyond it.

A Little Beyond Hartree-Fock

One way to include effects of interactions is to sum the following set of Feynman diagrams:

Fig.1: |χ22 (q, ω)|+|χ12 (q, ω)| in ladder approximation, for various q's.

They turn out to have algebraic structure that can be summed á la Dyson:

Part of it is plotted in Fig.1. (Click here for the fortran code and here for gnuplot.)

Within this approximation, we do find poles at low frequencies, presumably coming from spin waves. Also, a spin wave with enough energy (|ω| > E_gap) can dissociate into a pair of a Bloch-electron and a Bloch-hole. Thus it acquires finite-life time, with smearing of its pole expected.

© 2007 S. Yaida. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.