Friday, August 31, 2012

Getting the same answer from complementary numerical methods

Developing reliable numerical methods that can give meaningful and useful results for lattice models [e.g. the Hubbard, Heisenberg, and t-J models] of strongly correlated electrons is a challenging and tedious task. An important outcome is when complementary methods (including analytical methods) give the same result!

There is a nice Physical Review E article by Marcos Rigol, Tyler Bryant, and Rajiv Singh which considers the application of a new numerical linked cluster algorithm (NLC) method to the t-J model. To put nicely things in context they state
In spite of its simplicity, understanding finite-temperature thermodynamic properties of the t-J model has proven to be a very challenging task. Quantum Monte Carlo simulations suffer from severe sign problems, which become a major difficulty at low temperatures. The two general approaches that have been commonly used to study this model are [exact diagonalisation] ED and [high temperature expansions] HTE. ED studies in which one fully diagonalizes the t-J Hamiltonian are difficult since they can only be done for very small systems, as a consequence of which finite size effects are very large. A more efficient approach to this problem is the finite-temperature Lanczos method (FTLM), which has been developed by Jaklič and Prelovšek (JP). Within this approach the full thermodynamic trace is reduced by randomly sampling the eigenstates of the Hamiltonian. This allows one to study larger systems sizes in an unbiased way, but still finite size effects become relevant as the temperature is lowered.
The outstanding question concerning high temperature expansion (HTE) methods is whether they can give reliable results at the "low" temperatures relevant to experiments. Here it should be stressed that the energy scales t and J are of the order of 1000 K. On the other hand HTE and NLC have the distinct advantage that they are valid for the infinite lattice and do not suffer from finite size effects (a problem for FTLM and ED).

The figure below shows the very encouraging result that the complementary methods NLC and FTLM are in agreement for a calculation of the temperature dependence of the entropy, down to temperatures as low at about 0.1t, and for a range of dopings. 
This consistency increases the confidence of the reliability of both methods to describe the metallic phase of strongly correlated electron models. Some of the key underlying physics may be that the correlations are short-ranged in this regime of dopings and temperatures.

Two earlier posts considered the significance of FTLM results for understanding the metallic phase of cuprates at optimal doping. 


  1. Rajiv Singh asked me to post his comment:

    Thank you for the post. Perhaps one of Marcos' postdoc Ehsan can say something about how the method compares with QMC (even when there are no minus signs). It can be competitive or better than QMC if U is large enough.

    Another point, I would like to make is that compared with Exact Diagonalization (ED), NLC is particularly useful in 3D. Graphs are not that much harder to count as you go up in dimension, where as clusters with periodic boundary conditions needed
    in ED will have very small linear dimension associated with them and even if doable will bring a lot of artifact through the periodic boundary condition.

    One example is the problem of Quantum Spin Ice on a pyrochlore lattice. The smallest periodic cluster for ED that has all the symmetries of the pyrochlore lattice has 16 sites, which is difficult (though not impossible) to diagonalize.
    Most of the spin-space symmetries are absent in strong spin-orbit coupled systems. But, PBC will allow momentum as a good quantum number (for a 16 site cluster it only has 4 different values so will roughly reduce the Hilbert space by about a factor of 4). However, on this 16-site cluster, you only go about two neighbors in any direction before coming back to where you started. This is clearly very artificial.

    On the other hand, we recently carried out an NLC calculation for this problem in terms of clusters with complete tetrahedra to 4th order. The largest cluster had 13 sites. It gave reliable answers to thermodynamic properties down to a temperature of about J/2. The material Yb2Ti2O7 has largest exchange constant of about 2K, so the convergence was good down to 1K. In strong fields, NLC converges at all temperatures (down to T=0). That is because, the remainder (higher order terms) is high order in J/T and also in J/h. This covers a large range of parameters of interest to the experimental studies.

    A reference to our paper is:
    Vindication of Yb2Ti2O7 as a Model Exchange Quantum Spin Ice
    R. Applegate, N. R. Hayre, R. R. P. Singh, T. Lin, A. G. R. Day, and M. J. P. Gingras
    PRL 28 August 2012 (5 pages)

  2. Related to the topic of this thread and since Ross has mentioned NLC [it is probably better to call it NLCE (numerical linked cluster expansion) in the same spirit of HTE (high temperature expansion)] and FTLM as examples of numerical methods leading to the same results, I wanted to mention a recent study that involves NLCEs and quantum Monte Carlo (QMC) simulations on finite lattices. They were both used to explore thermodynamic properties and short-range correlations, of the single-band Fermi-Hubbard model on the square and honeycomb lattices. This work (arXiv:1206.0006), is a collaboration between myself, Marcos Rigol, his PhD student, Baoming Tang, and Thereza Paiva, and is part of a series of studies in which NLCEs have been used to explore finite temperature, and even ground state properties, of systems ranging from frustrated magnets to itinerant particles in optical lattices (see e.g., arXiv:1009.4700, arXiv:1104.5494, arXiv:1105.4147).

    We find that NLCE and QMC agree with each other very well for the two lattice geometries in the parameter regions that are accessible to both of them, despite the systematic errors that exist in QMC (e.g., finite size, or Trotter errors). Perhaps more interestingly, we find that the NLCE and QMC can be complementary numerical methods when it comes to Hubbard models; QMC runs into sampling problems as the ratio of the interaction strength over hopping amplitude, U/t, gets larger, while NLCE converges to lower temperatures in that regime. This can be understood as t^2/U becomes the relevant energy scale for large values of U/t, and NLCEs are very efficient up to where correlations (in this case, spin correlations) exceed the largest clusters considered. The opposite happens in the weak-coupling regime, where QMC is more efficient and NLCE's convergence temperature is typically higher. As a result of this, we have been able to, for example, map out the isentropic paths of the model at half filling in the temperature-interaction space for a wide range of interactions, up to three time the bandwidth (see Fig. 2 in arXiv:1206.0006). We also see that this agreement is not limited to half filling when we use a local density approximation to express different quantities of a systems that is subject to an external harmonic trapping potential, emulating the optical lattice experiments setup.

    Another feature of NLCE, and also an advantage over QMC-based methods, that I think is worth mentioning is that because of its exact diagonalization core, and therefore, full access to the partition function, properties such as the entropy or the specific heat can be calculated directly and without any need for numerical differentiation or integration. This has allowed us to obtain very accurate results for the specific heat of the strongly-coupled Fermi-Hubbard model in a wide temperature region (see Fig. 1 in arXiv:1204.1556).