In a previous report I found the U=0 energy spectrum for the transverse-field Hubbard model. Here I use it to compute the bare up-down spin susceptibility via
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| Fig. 1: Real (green) and imaginary (red) parts of the renormalized susceptibility are plotted for U=0,1,2,3, and t'=0, at momentum (3&pi/4,3&pi/4). The code used to produce these susceptibility plots is here. |
Diagramatically, this is represented by
The above expression is an exact form for the U=0 susceptibility. All well and good, but what we're really interested in is the behavior near the antiferromagnetic transition. To explore the susceptibility in that region, we incorporate the interaction term perturbatively. This is a delicate business, since perturbation theory breaks down near the phase boundary, but we nevertheless hope to obtain qualitatively accurate results by resumming certain classes of Feynman graphs. If we consider the up-down susceptibility, the most important easily-summable graphs are the ladders:
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| Fig. 2: Same as Fig.1, but evaluated at the ordering momentum q=(&pi,&pi). |
For our delta-function interaction, these terms form a geometric series, with sum
In Fig. 1, this renormalized susceptibility is shown, for a variety of values of U, but with fixed momentum q=(3&pi/4,3&pi/4). As U increases, a peak in the susceptibility separates out of the continuum and becomes an isolated pole. The presence of this resonance suggests that the system exhibits significant local spin ordering before the global antiferromagnetic transition.
In Fig. 2, the susceptibility is evaluated at the ordering momentum, q=(&pi,&pi). In this case, as U is increased to the phase boundary, spin wave poles do not emerge, but the susceptibility diverges near &omega=0, signalling the onset of critical behavior. In principle, critical exponents could be extracted from the behavior of the susceptibility in this region, but our badly behaved critical perturbation theory would be unlikely to produce trustworthy exponents.
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| Fig. 3: Dispersion relation for the pre-Goldstone modes, for U=3.1 (red) and for U=2 (green). |
It is natural to interpret the isolated pole in the susceptibility as the precursor to the Goldstone mode of the broken symmetry. Of course, the symmetry is not formally broken yet, but we would expect that the dispersion of this pole should be Goldstone-esque as the system moves towards the phase transition. Figure 3 shows the energy &omega of the resonance, as a function of q, for different choices of U (computed numerically, hence the "noise"). In both cases, the dispersion is roughly linear, as it should be for a Goldstone mode. The critical (U=3.1) mode has a smaller slope, and goes precisely to zero at small q, whereas the subcritical mode appears to have some non-zero mass. Both of these observations imply that exciting the Goldstone mode is "cheaper" close to the phase boundary. As always, the predictions of perturbation theory near the phase boundary should be taken with a grain of salt, however, so the details of the slope or mass calculations should not be trusted.
© 2007 D. Stanford. 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.