Comment on the paper "Quasi-particle approach for lattice Hamiltonians with large coordination numbers" by P. Navez, F. Queisser and R. Schützhold - J. Phys. A: Math. Theor. 47 225004 (2014)
aa r X i v : . [ c ond - m a t . o t h e r] S e p Comment on the paper “Quasi-particle approach for lattice Hamiltonians withlarge coordination numbers” by P. Navez, F. Queisser and R. Sch¨utzhold - J.Phys. A: Math. Theor. 47 225004 (2014)
D. Psiachos ∗ This comment regards a central aspect of the referred-to paper, the issue of convergence of thelarge coordination-number expansion. Perturbation expansions of expressions containing a largenumber of parameters are generally invalid due to the non-analyticity of the expanded expressions.I refer to recent work where these issues are analyzed and discussed in detail in relation to abenchmark example of a cluster model. As discussed therein, methods which are uncontrollable andfor which their convergence is not foreseeable are not only useless but can mislead, particularly ifmodels derived from them are used to interpret experiments.
The paper [1] aims to study some properties of generalized lattice-based systems: from ones with fewparameters e.g.
Heisenberg to more complex ones such as Bose-Hubbard (B-H) and Fermi-Hubbard (F-H).The central methodology used is a perturbation expansion in large coordination number (Z ≫
1) and it isapplied to compute density matrices, correlation functions and then dispersion relations. The results of thisexpansion are compared with some exact solutions in 1D, outside the supposed range of validity but mostimportantly, not for sufficiently-complex models with a large number of parameters such as the F-H and B-Hmodels.The models studied contain some or all of the following parameters: hopping, correlation, filling factor,Heisenberg exchange, and coordination number Z.However, it is well-known, and may easily be verified, that ad hoc perturbation expansions do not in generalwork for expressions which are non-analytic as the regions of convergence depend on the values taken byall the parameters present in the non-analytic portion. With an appropriate renormalization however, anew variable which does lead controllably to a convergent expansion may be defined, thereby giving clearconditions for the convergence as regards the relationship of all the parameters amongst themselves. Just asan example, in my recent work on assessing series expansions for a cluster-model benchmark system [3], in thiscase based on the two-band Anderson-Hubbard model, I have shown that for expressions containing multipleparameters, which for anything but the most trivial cases are most certainly non-analytic, a perturbationexpansion must be performed in terms of a new, renormalized variable which combines the parameters inthe non-analytic portion. Finding such a variable is very difficult, if not impossible, except for some simplesituations. Only then can a region of convergence be defined. Otherwise, the region of convergence of theexpansion will be dependent on the other parameters, in a way which is in practice unknown, thus renderingthe expansion unreliable. This argument equally holds for fixed order of expansion and increasing Z , e.g. for an ‘asymptotic’ expansion: it’s not clear how the validity is impacted by the values of other parametersin relation to Z .In Eq. C2 of [1] the authors present a comparison for a 1D (outside the supposed validity of the expansion)exact result - quantum Ising model - where the result has been shown to agree only in some limiting formsfor the parameters - not in general. For Eq. C1 (Heisenberg model), it is possible that owing to the fewparameters involved in the model that an agreement with the exact result is possible i.e. if they are notpresent in the non-analytic portion. However in order to be useful for replacing numerical calculations, sucha method must be demonstrated to converge in a determinate fashion, in a foreseeable region of parameterspace particularly for multi-parameter quantum-lattice models such as the B-H or F-H models treated inSecs. 6-7. For that to be able to be achieved in all parameter space, the renormalized expansion variablemust combine all those parameters found in the non-analytic portion of the full expression. The conclusionsreached in Ref. [3] regarding convergence in expressions with multiple parameters are sufficiently general soas to cover the work presented in the paper [1] even as they demonstrate their points using examples.**EDIT** The same argument holds for paper 2. ∗ Email: [email protected][1] P. Navez, F. Queisser and R. Sch¨utzhold.
J. Phys. A: Math. Theor. , 47:225004, 2014.[2] F. Queisser and K. V. Krutitsky and P. Navez and R. Sch¨utzhold.
Physical Review A , 89:033616, 2014.[3] D. Psiachos.