## Thursday, August 9, 2007

### Philip Phillips, UIUC: " Exact Integration of the High Energy Scale in Doped Mott Insulators"

Philip points out a problem with the naive procedures used to integrate out high energy degrees of freeedom.
As an example, in eliminating the upper Hubbard band in order to
derive the low energy spectra of the large-U Hubbard model,one might miss the enhanced spectral weight for the addition spectrum of holes in the lower Hubbard band. (Comment: a fact which is missed in simple Hartree Fock theory.)

In his talk, Philips sets out to preserve the "2X" sum rule (i.e. the holes' spectral weight is -twice- their doping concentration, as hole doping pulls states down from the upper Hubbard band).

Philips' method of choice is an introduction of addtional fields, a fermion field which counts the high energy doublons (sites with two electrons), and a constraint field -phi- which projects the enlarged Hilbert space back to the original electrons states.
The result is a formally quadratic Lagrangian, with matrix (and space-time dependent) coupling parameters which is a starting point for a saddle point expansion. Its saddle point includes the t-J model terms and is argued to be a better description of the Hubbard model's low energy spectral weight. Predictions were made about a second dispersing peak which may be observed in the ARPES data.
Time limitations have restricted questions to a minimum, but Patrick Lee commented that
although the effective Lagrangian is formally correct, its fluctuations are large in the large U/t limit.

I argued that the traditional renormalization procedures (Brillouin-Wigner perturbative expansion, Real space Contractor Renormalization (CORE)) are somehat simpler and that the t-J model correctly captures the low energy spectrum. However in a later discussion with Philip and Patrick, Philip's approach was understood as an attempt to simplify the calculation of
quasiparticle renormalization and the intermediate energy scale excitations.

Piers Coleman said...

Patrick Lee made the comment that the constraint field in Philip Phillips approach has wild fluctuations, and therefore is not useful as the basis for a low energy theory. I believe that Patrick is incorrect in this respect. The bosonic constraint field in this approach plays a very similar role to the constraint field in the Lee and Wen SU(2) approach to the t-J model - and indeed, in the large N limit plays the role of eliminating s-wave pairs from the Hilbert space.
Indeed, for the negative U Hubbard model, this field will almost certainly develop an expectation value - to describe the sc ground-state. In the positive U model, it will adjust to prevent any s-wave or extended s-wave pairing from development.

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We expand on our earlier work (cond-mat/0612130, Phys. Rev. Lett. {\bf 99}, 46404 (2007)) in which we constructed the exact low-energy theory of a doped Mott insulator by explicitly integrating (rather than projecting) out the degrees of freedom far away from the chemical potential.

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