Monday, March 4, 2013

Interplay of dynamical and spatial fluctuations near the Mott transition

There is a nice preprint The Crossover from a Bad Metal to a Frustrated Mott Insulator by Rajarshi Tiwari and Pinaki Majumdar

They study my favourite Hubbard model: the half-filled model on an anisotropic triangular lattice, within a new approximation scheme.
Basically, they start with a functional integral representation and ignore the dynamical fluctuations in the local magnetisation. One is then left with calculated the electronic spectrum for an inhomogeneous spin distribution and averaging then averaging over these with the relevant Boltzmann weights. This has the significant computational/technical advantage that the calculation is a classical Monte Carlo simulation.
Hence, it treats spatial spin fluctuations exactly while neglecting dynamical fluctuations. 

This is a nice study because it complements the approximation of dynamical mean-theory  (DMFT): it ignores spatial fluctuations but treats dynamical fluctuations exactly.

The calculation captures some of the important properties of the model: the Mott transition, a bad metal, and a possible pseudogap phase.

This shows how an anisotropic pseudogap can arise in the model due to short-range antiferromagnetic spin fluctuations (clearly shown in Figure 5 of the paper, reproduced below). 

However, as I would expect, this approximation cannot capture some of the key physics that DMFT does: the co-existence of Hubbard bands and a Fermi liquid. This difference is clearly seen in the optical conductivity calculated by the two different methods.

There must be some connection with old studies [motivated by the cuprates] of Schmalian, Pines, and Stojkovic, of electrons coupled to static spin fluctuations with finite-range correlations [see e.g. this PRB].  

This combined importance of both dynamical and spatial fluctuations highlights to me the importance of a recent study by Jure Kokalj and I, which treated them on the same footing by using the finite temperature Lanczos method on small lattices. 


  1. For U less than the critical U for the Mott transition, it is possible to observe a pseudogap in two dimensions if the antiferromagnetic correlation length becomes larger than the thermal de Broglie wavelength. A pseudogap is also possible at U larger than the critical U for Mott transition, but that is a different mechanism. It is not clear for me what is the dominant mechanism in this paper.

    Also, by the way, another difference between the CDMFT results and these is the absence of a critical point. The critical U is also smaller by about a factor of two.

  2. Both spatial and dynamical fluctuations
    are important in understanding the Mott
    transition in frustrated systems, particularly
    at low temperature. DMFT, in its single site
    variant, and the present method approach the
    problem from two different ends.

    The present approach considers thermal
    fluctuations `exactly', with the caveat that
    at low T these are only fluctuations about the
    unrestricted Hartree-Fock state! At higher
    temperature, however, where `zero point effects'
    are no longer dominant, the spatial thermal
    fluctuations should well approximate the full

    DMFT retains `all local quantum fluctuations' but is oblivious to spatial correlations. They get built in slowly as one moves to the cluster variants, but these calculations are probably not sophisticated enough yet to capture complex magnetic order. Again, at high temperature,
    where correlation lengths are short, the DMFT
    answer, with proper choice of the bath, should
    be a good approximant.

    CDMFT, or DCA, pushed to bigger sizes would be
    one way to go.. another would be to build in
    dynamical fluctuations on the static auxiliary
    field scheme in a hierarchical spirit. Easier
    said ..

    Quick comment on Prof. Tremblay's remark:

    (i) Within our scheme pseudogaps arise due to
    electron coupling to moderately large but spatially
    `disordered' local moments. The same basic
    mechanism works both below and above U_c.
    On the metallic side the moments themselves are
    thermally generated and only short range correlated.
    On the insulating side the moments exist and are
    ordered at T=0 (and create a gap). Their thermal
    disordering can create a PG even below T_c.
    Too large a moment, as in the large U insulator,
    would sustain a gap even after disordering.

    (ii) The U_c differs from CDMFT. True! So does
    the CDMFT result from the single site DMFT. I guess
    one slowly converges towards the truth..