Legacy Bosonization

Developing an approach to generic condensed matter problems that would be as powerful and versatile as 1d bosonization has long remained as desirable, as it has been elusive. The vigorous early studies in that direction [19–21] were followed by a long period of oblivion, although there has been some revival of interest, as of lately [22]. Thus far, however, concrete predictions of this technique amounted to reproducing the classic FL results [19] or, else, delivering some novel ones which appear to be quite similar to those obtained by means of the eikonal approach.

At the practical level, the simplest version of the bosonization technique is described as a triangulation procedure in the course of which the FS gets divided onto a large number of ’patches’ whose size decreases upon renormalization. The idea is that a predominantly forward scattering makes physics quasi-1d and subject to the similar (albeit, approximate in d > 1) Ward identities [23]. One might then hope that this would suffice for permitting the use of the 1d rules of substituting bosonic (’plasmon’) density modes for fermion bilinears at small energies/momenta.

The equivalent bosonic action is formulated in terms of the phase variable φ(τ, r, n) labeled by its location on the surface of constant Fermi energy traced by the unit normal n (or the corresponding d−1 angular parameters)

where D(τ, r) is the Fourier transform of (1) and the sum is over the FS patches of a progressively decreasing size.

The proposed ’ad hoc’ construction of the single-particle operator [19]

ignores any operator-valued factors a la Klein that in 1d make the thus-obtained fermion operators obey the standard commutation relations at different FS points. Therefore, while adequately reproducing the low-energy/momentum FS hydrodynamics at the level of (charge/spin)density response functions the bosonization recipe (18) apriori may not be sufficient when it comes to single-particle amplitudes.

Nonetheless, the single-particle propagator obtained with the use of (18) is given by the expression

where the brackets stand for the average over the phase fluctuations governed by the Gaussian action (17).

In the non-interacting case the corresponding integral is logarithmic, thus resulting in the angular expansion of the free propagator (3) with the dispersion linearized near the FS

Neglecting any effects of the FS curvature one obtains a formula reminiscent of the eikonal propagator (9)

Although the original bosonization approach appears to be rather similar to the eikonal technique, it can be further improved. To that end, its more sophisticated version (see [21] and its recent redux [22]) can be formulated in terms of a path integral over the Boltzmann distribution function which plays the role of a collective bosonic field variable. The rigorous formulation of this approach makes use of the Costant-Kirillov method of quantization on the coajoint orbits of the phase space volumepreserving diffeomorphisms whose generators obey W∞ algebra.

However, while potentially providing a systematic way to refine (21) by taking into account the non-linear terms (higher derivatives of φ(τ, r, n)) in the effective hydrodynamics, this technique has not yet demonstrated its full potential. Same can be said about a very different scheme (which is formally exact by design, too) that exploits intrinsic supersymmetry by introducing ancillary ghost fermions [25].

Albeit still waiting to be explored, neither way of improving on the lowest-order eikonal/bosonization results is expected to be an easy task. In that regard, the recent (largely, verbal) claim [17] of exactness of the asymptotic (21) seems much too simple to be true.

Author:

(1) D. V. Khveshchenko, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599.

This paper is available on arxiv under CC BY 4.0 DEED license.