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arXiv:1901.00029 · Physical review. B./Physical review. B

Multivaluedness of the Luttinger-Ward functional in the fermionic and bosonic system with replicas

Aaram J. Kim, Vincent Sacksteder

Published Mar 31, 2020• 22 citations
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Abstract

We study the properties of the Luttinger-Ward functional (LWF) in a simplified Hubbard-type model without time or spatial dimensions, but with $N$ identical replicas located on a single site. The simplicity of this $(0+0)d$ model permits an exact solution for all $N$ and for both bosonic and fermionic statistics. We show that fermionic statistics are directly linked to the fact that multiple values of the noninteracting Green's function ${G}_{0}$ map to the same value of the interacting Green's function $G$; that is, the mapping ${G}_{0}\ensuremath{\mapsto}G$ is noninjective. This implies that with fermionic statistics the $(0+0)d\phantom{\rule{4pt}{0ex}}N$-replica model has a multiply valued LWF. The number of LWF values in the fermionic model increases proportionally to the number of replicas $N$, while in the bosonic model the LWF has a single value regardless of $N$. We also discuss the formal connection between the $N$-replica model and the $(0+1)d$ Hubbard atom which was used in previous studies of LWF's multivaluedness.

DOI
10.1103/physrevb.101.115146

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•Jan 1, 2002•0 citations
We study the properties of the Luttinger-Ward functional (LWF) in a simplified Hubbard-type model without time or spatial dimensions, but with NNN identical replicas located on a single site. The simplicity of this (0+0)d(0+0)d(0+0)d model permits an exact solution for all NNN and for both bosonic and fermionic statistics. We show that fermionic statistics are directly linked to the fact that multiple values of the noninteracting Green's function G0{G}_{0}G0​ map to the same value of the interacting Green's function GGG; that is, the mapping G0\ensuremath↦G{G}_{0}\ensuremath{\mapsto}GG0​\ensuremath↦G is noninjective. This implies that with fermionic statistics the (0+0)dN(0+0)d\phantom{\rule{4pt}{0ex}}N(0+0)dN-replica model has a multiply valued LWF. The number of LWF values in the fermionic model increases proportionally to the number of replicas NNN, while in the bosonic model the LWF has a single value regardless of NNN. We also discuss the formal connection between the NNN-replica model and the (0+1)d(0+1)d(0+1)d Hubbard atom which was used in previous studies of LWF's multivaluedness.
The Luttinger-Ward functional (LWF) has been a starting point for conserving approximations in many-body physics for 50 years. The recent discoveries of its multivaluedness and the associated divergence of the two-particle irreducible vertex function $\mathrm{\ensuremath{\Gamma}}$ have revealed an inherent limitation of this approach. Here we demonstrate how these undesirable properties of the LWF can lead to a failure of computational methods based on an approximation of the LWF. We apply the nested cluster scheme (NCS) to the Hubbard model and observe the existence of an additional stationary point of the self-consistent equations, associated with an unphysical branch of the LWF. In the strongly correlated regime, starting with the first divergence of $\mathrm{\ensuremath{\Gamma}}$, this unphysical stationary point becomes attractive in the standard iterative technique used to solve DMFT. This leads to an incorrect solution, even in the large cluster size limit, for which we discuss diagnostics.
The Luttinger-Ward functional $\mathrm{\ensuremath{\Phi}}[\mathbf{G}]$, which expresses the thermodynamic grand potential in terms of the interacting single-particle Green's function G\mathbf{G}G, is found to be ill defined for fermionic models with the Hubbard on-site interaction. In particular, we show that the self-energy $\mathbf{\ensuremath{\Sigma}}[\mathbf{G}]\ensuremath{\propto}\ensuremath{\delta}\mathrm{\ensuremath{\Phi}}[\mathbf{G}]/\ensuremath{\delta}\mathbf{G}$ is not a single-valued functional of G\mathbf{G}G: in addition to the physical solution for $\mathbf{\ensuremath{\Sigma}}[\mathbf{G}]$, there exists at least one qualitatively distinct unphysical branch. This result is demonstrated for several models: the Hubbard atom, the Anderson impurity model, and the full two-dimensional Hubbard model. Despite this pathology, the skeleton Feynman diagrammatic series for $\mathbf{\ensuremath{\Sigma}}$ in terms of G\mathbf{G}G is found to converge at least for moderately low temperatures. However, at strong interactions, its convergence is to the unphysical branch. This reveals a new scenario of breaking down of diagrammatic expansions. In contrast, the bare series in terms of the noninteracting Green's function G0{\mathbf{G}}_{0}G0​ converges to the correct physical branch of $\mathbf{\ensuremath{\Sigma}}$ in all cases currently accessible by diagrammatic Monte Carlo calculations. In addition to their conceptual importance, these observations have important implications for techniques based on the explicit summation of the diagrammatic series.
We consider a number of questions regarding the Luttinger-Ward functional and the many-body perturbation series expansion of the proper self-energy Σ(k;z)Σ(\mathbf{k};z)Σ(k;z) specific to uniform ground states (ensemble of states) of interacting fermion systems in terms of skeleton self-energy diagrams and the interacting Green function G(k;z)G(\mathbf{k};z)G(k;z). Utilising a link between the latter series expansion and the classical moment problem (of the Hamburger type), along with the associated continued-fraction expansion, we reaffirm our earlier observation (2007) that for lattice models of fermions interacting through short-range two-body potentials (i.e. for Hubbard-like models) this series is uniformly convergent for almost all wave vectors k\mathbf{k}k and complex energies zzz. The limit of this series is unique. We inquire into the reasons underlying the contrary observation by Kozik et al. (2015) regarding skeleton-diagrammatic perturbation series expansion of ΣΣΣ. In doing so, we make a number of observations of general interest. Chief amongst these, we observe that contrary to general belief thermal correlation functions calculated in the energy domain (i.e. over the set of the relevant Matsubara frequencies) are generally unreliable, in contrast to those calculated in the imaginary-time domain. In an appendix we present a symbolic computational formalism and short programs for singling out topologically distinct proper self-energy diagrams that are algebraically equivalent up to determinate multiplicative constants. This complements the work presented in our previous publication (2019). [Abridged abstract]
We analyze the original proof by Luttinger and Ward of the Luttinger theorem, according to which for uniform ground states of systems of (interacting) fermions, which may be metallic or insulating, the number of k points corresponding to non-negative values of G_s(k;mu) is equal to the total number of particles with spin index s in these ground states. Here G_s(k;mu) is the single-particle Green function of particles with spin index s at the chemical potential mu. For the cases where the two-body interaction potential is short-range, and in particular for lattice models, we explicitly demonstrate that this theorem is unconditionally valid, irrespective of the strength of the bare interaction potential. We arrive at this conclusion by amongst other things demonstrating that the perturbation series expansion for self-energy in terms of skeleton diagrams, as encountered in the proof of the Luttinger-Ward identity, is uniformly convergent for almost all momenta and energies. We further investigate the mechanisms underlying some reported instances of failure of the Luttinger theorem. With one exception, for all the cases considered in this paper, we show that the apparent failures of the Luttinger theorem can be attributed either to shortcomings of the employed single-particle Green functions or to misapplication of this theorem. The one exceptional case brings to light the possibility of a genuine failure of the Luttinger theorem for insulating ground states, which we show to be brought about by a false limit that in principle can be reached on taking the zero-temperature limit without the value of mu coinciding with the zero-temperature limit of the chemical potential satisfying the equation of state at finite temperatures; no such ambiguity can arise for metallic states.
The local two-particle Green's function and its generalized susceptibility are central building blocks in state-of-the-art theoretical descriptions of strongly correlated many-electron systems. The unexpected discovery of singular points in the generalized susceptibility, resulting in divergences of the two-particle irreducible vertex function and in multivaluedness of the Luttinger-Ward functional, has therefore attracted considerable interest. Unfortunately, almost no analytical results for irreducible vertices are available for correlated electrons. Here, the authors fill this gap by presenting fully analytical expressions for all irreducible vertex function of a simple, yet prototypical, correlated model, the single-orbital Hubbard atom. This gives highly relevant insights into the energy dependence of these functions.
The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low-energy spectrum of noninteracting fermions is identical to the one for a harmonic chain. This allows us to describe interacting fermions as a system of coupled oscillators. The exact solution of the Tomonaga-Luttinger model is discussed and results for the momentum distribution and spectral functions are presented. The general Luttinger liquid phenomenology is briefly discussed and various attempts at the experimental verification of the theoretical predictions are examined. Extended defects in semiconductors might also be candidate systems.