We prove the convergence of a simplified cluster/Mayer expansion at one energy scale for three space-time dimensional many Fermion systems. The bounds are uniform in the scale. We iterate them to show that the sum of all diagrams that contain no two or four-legged subdiagrams converges. Our results are suited to a multiscale construction of the full system. Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Schweizerischer Nationalfonds zur Forderung der wissenschaftlichen Forschung y Research supported in part by the Forschungsinstitut fur Mathematik, Zurich xI Introduction In this paper we consider many Fermion systems formally characterized by the effective potential G(/ e ; ¯ / e ) = log 1 Z Z e \\GammaV (/+/ e ; ¯ /+ ¯ / e ) d¯C (/; ¯ /) for the external fields / e ; ¯ / e . Here, d¯C (/; ¯ /) is the fermionic Gaussian measure in the Grassmann variables f/(¸); ¯ /(¸) j ¸ 2 IR \\Theta IR d \\Theta ...
We prove the convergence of a simplified cluster/Mayer expansion at one energy scale for three space-time dimensional many Fermion systems. The bounds are uniform in the scale. We iterate them to show that the sum of all diagrams that contain no two or four-legged subdiagrams converges. Our results are suited to a multiscale construction of the full system. Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Schweizerischer Nationalfonds zur Forderung der wissenschaftlichen Forschung y Research supported in part by the Forschungsinstitut fur Mathematik, Zurich xI Introduction In this paper we consider many Fermion systems formally characterized by the effective potential G(/ e ; ¯ / e ) = log 1 Z Z e \GammaV (/+/ e ; ¯ /+ ¯ / e ) d¯C (/; ¯ /) for the external fields / e ; ¯ / e . Here, d¯C (/; ¯ /) is the fermionic Gaussian measure in the Grassmann variables f/(¸); ¯ /(¸) j ¸ 2 IR \Theta IR d \Theta ...
In this paper we complete the first step, namely the uniform bound on completely convergent contributions, towards proving that a three dimensional interacting system of Fermions is a Fermi liquid in the sense of Salmhofer. The analysis relies on a direct space decomposition of the propagator, on a bosonic multiscale cluster expansion and on the Hadamard inequality, rather than on a Fermionic expansion and an angular analysis in momentum space, as was used in the recent proof by two of us of Salmhofer’s criterion in two dimensions.
Abstract We provide a mathematically rigorous Keldysh functional integral for fermionic quantum field theories. We show convergence of a discrete-time Grassmann Gaussian integral representation in the time-continuum limit under very general hypotheses. We also prove analyticity of the effective action and explicit bounds for the truncated (connected) expectation values γm,mˉc γ m , m ¯ c of the non-equilibrium system. These bounds imply clustering with a summable decay in the thermodynamic limit, provided these properties hold at time zero, and provided that the determinant bound δC δ C and decay constant αC α C of the fermionic Keldysh covariance are bounded uniformly in the volume. We then give bounds for these constants and show that uniformity in the volume indeed holds for a general class of systems. Finally we show that in the setting of dissipative quantum systems, these bounds are not necessarily restricted to short times.
. We derive Ward identities, to all orders of perturbation theory, in many Fermion systems with short range interactions and broken U(1) number symmetry. They imply that the system is superrenormalizable in the Goldstone boson regime. Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Schweizerischer Nationalfonds zur Forderung der wissenschaftlichen Forschung y Research supported in part by the Forschungsinstitut fur Mathematik, Zurich 1 xI Introduction We consider a many Fermion system, in dimension d 2 , formally characterized by the (renormalized) generating functional S(OE; OE) = log 1 Z Z e [ OE/]+[ /OE] e A(/; /) Q k;oe d/ k;oe d / k;oe (I:1) for the connected Euclidean Green's functions, where the action A(/; /) = \\GammaV (/; /) \\Gamma ffi(; ) Z dk / k / k \\Gamma Z dk \\Gamma ik 0 e(k) \\Delta / k / k (1:2) and V(/; /) = 2 Z 4 Q i=1 dk i (2) d+1 ffi(k 1 +k2 \\Gamma...
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) 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). 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 and complex energies z. 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 Σ. 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 obtain an explicit expression for the multipoint energy correlations of a non-solvable two-dimensional Ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength λ, in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one. Our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory, which generalizes the one originally used by Schultz, Mattis, and Lieb for the nearest neighbor Ising model. The interacting model is then analyzed by a multiscale method first proposed by Pinson and Spencer. If the lattice spacing is finite, then the correlations cannot be computed in closed form: rather, they are expressed in terms of infinite, convergent, power series in λ. In the scaling limit, these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable Ising model, up to a finite renormalization of the parameters. Explicit bounds on the speed of convergence to the scaling limit are derived.
The ground-state energy and low-energy excitations of single-particle character of an infinite medium of fermions are discussed with the aid of time-dependent Green's functions, which are convenient generalizations of the exact particle correlation functions in the ground state of N particles. The power series development for the one- and two-particle functions, under the restriction to two-body forces, is derived and described by means of Feynman diagrams. The derivation of the linked-cluster expansion for the energy then follows immediately. The equivalence to previous versions is established. The one-particle function is examined in particular detail, and it is shown that the poles of its space-time Fourier transform studied as a function of the energy variable, for fixed momentum, determined the (N+1)-particle and ($N\ensuremath{-}1$)-particle excited states which have single-particle character. For a reasonable assumption about the full spectrum of excited states, it is found that for the interacting system, single-particle excitations with a real energy occur only at the Fermi momentum. It is pointed out that the corresponding energy, termed the perturbed Fermi energy, equals the binding energy per particle in the ground state of N particles for a saturating system at equilibrium density.It is shown, finally, that the entire structure of the theory may be carried over to the case of finite temperature, requiring only a redefinition of the Green's functions. The analogy is constructed from a discussion of the internal energy.