# Connected determinant diagrammatic Monte Carlo: polynomial complexity despite fermionic sign

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## Generalization to correlation functions and to the fully-dressed diagrammatic scheme

In this section, we show how it is possible to generalize the large-order behavior computation to the correlation functions, and how to consider more general diagrammatic schemes.

### Borel summability of the self-energies in the ladder expansion

The computation for the pressure is not generalizable as it is to correlation functions, as they are expressed as ratio of partition-like functions. The strategy is then very simple: we compute the large-order behavior for each partition-like function, and we deduce the large-order behavior of the correlation function. In this section we are interested in bounding the large-order behavior, in the next section we will make explicit predictions for it. In order to manipulate such ratios of partition functions, which are given by asymptotic series which respect certain bounds, it is very useful to introduce the concept of Gevrey asymptotic series: (Gevrey asymptotic series): Let A > 0. A formal power seriesP 1 n = 0 z n f n is Gevrey asymptotic of type (5;A) to the function f(z) if there exists R > 0 and > 0 such that f(z) is analytic for z 2 WR := fz 2 C j 0 < jzj < R; jarg zj < =10 + g and such that for every 0 < A < 1, there exists C < 1 such that for every N 2 N and z 2 WR .

#### Alternative view: discontinuities near the origin

It is a well-known fact that for analytic functions with non-zero radius of convergence of the Taylor expansion at the origin, the large-order behavior of the coefficients is determined by the properties of the singularity nearest to the point of expansion. Having a zero radius of convergence means that there are singularities exactly at zero distance from the point of expansion, otherwise it is straightforward to show that the radius of convergence would be finite. We have extended the Borel transform B(z) of pladd(z) to z 2 C n fz 2 C j jzj Aladd; jarg zj = 4=5g. As before, we write (in this section we drop the b1 and b2 parameters of the Borel transform for simplicity).

Discussion of the bold scheme

There is no difficulty in repeating the analysis of the previous section for general partially dressed schemes, like the semi-bold scheme introduced in . However, for the bold scheme introduced in Section 1.3, the analysis cannot be applied without changes, as there are singularities coming from the fact that the shifted action itself is singular at the origin, as it is built self-consistently with the selfenergy and pair self which are singular at the origin31. It is then clear that in the fully-bold case we cannot expect the same level of rigour as for other schemes, but we will try nevertheless to give rather convincing arguments.
First of all, we need to define the bold scheme in a more formal way than in Section 1.3. We can define these series in a completely formal way starting from the bare expansion32. We remind that

Large-field behavior of the determinant

It is well known that the large external potential limit in quantum mechanics is given by a quasi-local approximation. Indeed, as the wavelength of the particle is proportional the inverse of the square root of the potential depth, the particle experience an essentially flat potential over one wavelength. The integration over fermions gives the partition function of a (one-particle) imaginary-time quantum-mechanical system with a (pairing) external potential, with anti-periodic boundary conditions in the imaginary-time direction. It seems clear physically that we can apply therefore the quasi-local (or Thomas-Fermi) approximation. We just need now to justify this physical picture.

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Introduction
Overview
I Large-order behavior and resummation for the unitary Fermi gas
1 Introduction
1.1 The unitary Fermi gas
1.2 The unitary Fermi gas as the continuum limit of lattice models
1.3 Diagrammatic dressing: bare, ladder, and bold expansions
2 Summary of main results
3 Large-order behavior for the pressure in the ladder scheme
3.1 Shifted action for the ladder scheme
3.2 Obtaining the large-order behavior for the pressure from the partition function
3.3 Integration over fermions
3.4 Saddle point for the large-order behavior of the determinant
3.5 Functional integration in the large-n limit
4 Analytic reconstruction by conformal-Borel transform
4.1 Unicity of Taylor coefficients
4.2 Borel summability and conformal mapping
5 Generalizations
5.1 Borel summability of the self-energies in the ladder expansion
5.2 Alternative view: discontinuities near the origin
5.3 Discussion of the bold scheme
6 Numerical results
7 Conclusions and outlook
Appendices of Part I
A Shifted action for the ladder scheme
B Interchanging thermodynamic and large-order limit
C Renormalized Fredholm determinants
D Large-field behavior of the determinant
E No imaginary-time dependence of the instanton
F Sobolev bound
G Variational principle
H High and low temperature limits of the action functional
I Symmetric decreasing rearrangement
J Gaussian zero modes
K Gaussian integration
L Bound on derivatives
M Unicity of analytic continuation
N Ramis’s theorem
O Consistency condition for the maximal analytic extension
P Construction of the conformal map
Q Some properties of Gevrey asymptotic series
R Dispersion relation
S Pair propagator in terms of the fields
T Generalized Pauli formulas
U Numerical computation of the minimum of the action
V Legendre-Bateman expansion
W Discontinuities of the bold self-energies
II Connected determinant diagrammatic Monte Carlo: polynomial complexity despite fermionic sign
Article 1: Determinant Diagrammatic Monte Carlo in the thermodynamic limit
Article 2: Polynomial complexity despite the fermionic sign
Complements
1 The recursive formula for many-variable formal power series
2 Example: two-body interactions with a linear quadratic shift .
2.1 Calculation of aE(V )
2.2 Monte Carlo integration and computational cost of the non-deterministic part
2.3 Generalizations
2.4 Computation of physical correlation functions
3 Polynomial-time scaling
Conclusion and outlook
Appendices of Part II
A Computational cost of the recursive formula
B Computational cost of the determinants
C Hybrid sampling
D Binary representation
III Multivaluedness of the Luttinger-Ward functional and applicability of dressed diagrammatic schemes
1 Introduction and overview
2 The dangers of dressing
2.1 The formal definition of the bold series
2.2 The Kozik-Ferrero-Georges branch of the Luttinger-Ward functional
2.3 Misleading convergence of the bold scheme
3 An insightful model
3.1 Feynman diagrams definition
3.2 Grassmann integral representation
3.3 The KFG branch of the toy model
3.4 Misleading convergence of the bold scheme within the toy model
4 Semi-bold scheme: partial dressing with no misleading convergence
4.1 Shifted-action expansion
4.2 Semi-bold scheme
5 Applicability of the bold scheme
6 Conclusions and outlook
Article 3: Skeleton series and multivaluedness of the self-energy
functional in zero space-time dimensions
Article 4: Shifted-action expansion and applicability of dressed diagrammatic schemes
Bibliography

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